Boomerang sequences

Let's say that a “boomerang sequence” is a sequence returning unchanged after one or more mathematical operations have been performed on its terms (a "boomerang sequence" is not a "boomerang number").

We met already two such sequences, see here the sequences A369603 and A369604, recently submitted to the OEIS and computed by Giorgos Kalogeropoulos.

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Feb 2, 2024

Update : two new "boomerang seqs" at the very bottom of this page

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What about this one?

« Multiply each digit d of S by the number to which d belongs: the sequence S remains identical to itself if we follow each multiplication with a comma. »

S starts like this:

S = 0, 1, 12, 24, 48, 96, 192, 384, 864, 576, 192, 1728, 384, 1152, 3072, 1536, 6912, ...

Check

a(1) = 0, which multiplied by 0 gives 0

a(2) = 1, which multiplied by 1 gives 1

a(3) = 12

1st digit is 1, which multiplied by 12 gives 12

2nd digit is 2, which multiplied by 12 gives 24

a(4) = 24

1st digit is 2, which multiplied by 24 gives 48

2nd digit is 4, which multiplied by 24 gives 96

a(5) = 48

1st digit is 4, which multiplied by 48 gives 192

2nd digit is 8, which multiplied by 48 gives 384

a(6) = 96

1st digit is 9, which multiplied by 96 gives 864

2nd digit is 6, which multiplied by 96 gives 576

... etc.

We see that the above last yellow terms reproduce S.

S is the lexicographycally earliest nontrivial seq with this property: if we try for a(3) the integers 1, 10 or 11, we respectively get:

S = 0, 1, 1, 1, 1, 1, 1, 1, 1, ...

S = 0, 1, 10, 0, 0, 0, 0, 0, ...

S = 0, 1, 11, 1, 1, 1, 1, 1, 1, ...

Jean-Marc Falcoz has extended S to 14937 terms (they are all even, of course – except a(1) –, and 1403 zeros are present so far):

S = 0, 1, 12, 24, 48, 96, 192, 384, 864, 576, 192, 1728, 384, 1152, 3072, 1536, 6912, 5184, 3456, 2880, 4032, 3456, 192, 1728, 384, 1728, 12096, 3456, 13824, 1152, 3072, 1536, 1152, 1152, 5760, 2304, 9216, 0, 21504, 6144, 1536, 7680, 4608, 9216, 41472, 62208, 6912, 13824, 25920, 5184, 41472, 20736, 10368, 13824, 17280, 20736, 5760, 23040, 23040, 0, 16128, 0, 12096, 8064, 10368, 13824, 17280, 20736, 192, 1728, 384, 1728, 12096, 3456, 13824, 1152, 3072, 1536, 1728, 12096, 3456, 13824, 12096, 24192, 0, 108864, 72576, 10368, 13824, 17280, 20736, 13824, 41472, 110592, 27648, 55296, 1152, 1152, 5760, 2304, 9216, 0, 21504, 6144, 1536, 7680, 4608, 9216, 1152, 1152, 5760, 2304, 1152, 1152, 5760, 2304, 28800, 40320, 34560, 0, 4608, 6912, 0, 9216, 82944, 18432, 9216, 55296, 43008, 21504, 107520, 0, 86016, 36864, 6144, 24576, 24576, 1536, 7680, 4608, 9216, 53760, 46080, 61440, 0, 18432, 27648, 0, 36864, 82944, 18432, 9216, 55296, 165888, 41472, 165888, 290304, 82944, 373248, 124416, 124416, 0, 497664, 41472, 62208, 6912, 13824, 13824, 41472, 110592, 27648, 55296, 51840, 129600, 233280, 51840, 0, 25920, 5184, 41472, 20736, 165888, 41472, 165888, 290304, 82944, 41472, 0, 145152, 62208, 124416, 10368, 0, 31104, 62208, 82944, 13824, 41472, 110592, 27648, 55296, 17280, 120960, 34560, 138240, 0, 41472, 0, 145152, 62208, 124416, 28800, 40320, 34560, 0, 46080, 69120, 0, 92160, 0, 46080, 69120, 0, 92160, 0, 16128, 96768, 16128, 32256, 129024, 12096, 24192, 0, 108864, 72576, 64512, 0, 48384, 32256, 10368, 0, 31104, 62208, 82944, 13824, 41472, 110592, 27648, 55296, 17280, 120960, 34560, 138240, 0, 41472, 0, 145152, 62208, 124416, 192, 1728, 384, 1728, 12096, 3456, 13824, 1152, 3072, 1536, 1728, 12096, 3456, 13824, 12096, 24192, 0, 108864, 72576, 10368, 13824, 17280, 20736, 13824, 41472, 110592, 27648, 55296, 1152, 1152, 5760, 2304, 9216, 0, 21504, 6144, 1536, 7680, 4608, 9216, 1728, 12096, 3456, 13824, 12096, 24192, 0, 108864, 72576, 10368, 13824, 17280, 20736, 13824, 41472, 110592, 27648, 55296, 12096, 24192, 0, 108864, 72576, 48384, 96768, 24192, 217728, 48384, 108864, 0, 870912, 870912, 653184, 435456, 508032, 145152, 362880, 508032, 435456, 10368, 0, 31104, 62208, 82944, 13824, 41472, 110592, 27648, 55296, 17280, 120960, 34560, 138240, 0, 41472, 0, 145152, 62208, 124416, 13824, 41472, 110592, 27648, 55296, 165888, 41472, 165888, 290304, 82944, 110592, 110592, 0, 552960, 995328, 221184, 55296, 193536, 165888, 110592, 221184, 276480, 276480, 110592, 497664, 331776, 1152, 1152, 5760, 2304, 1152, 1152, 5760, 2304, 28800, 40320, 34560, 0, 4608, 6912, 0, 9216, 82944, 18432, 9216, 55296, 43008, 21504, 107520, 0, 86016, 36864, 6144, 24576, 24576, 1536, 7680, 4608, 9216, 53760, 46080, 61440, 0, 18432, 27648, 0, 36864, 82944, 18432, 9216, 55296, 1152, 1152, 5760, 2304, 1152, 1152, 5760, 2304, 28800, 40320, 34560, 0, 4608, 6912, 0, 9216, 1152, 1152, 5760, 2304, 1152, 1152, 5760, 2304, 28800, 40320, 34560, 0, 4608, 6912, 0, 9216, 57600, 230400, 230400, 0, 0, 161280, 0, 120960, 80640, 0, 103680, 138240, 172800, 207360, 0, 18432, 27648, 0, 36864, 41472, 62208, 6912, 13824, 82944, 18432, 9216, 55296, 663552, 165888, 746496, 331776, 331776, 18432, 147456, 73728, 55296, 36864, 82944, 18432, 9216, 55296, 276480, 276480, 110592, 497664, 331776, 172032, 129024, 0, 0, 344064, 43008, 21504, 107520, 0, 86016, 107520, 0, 752640, 537600, 215040, 0, 688128, 516096, 0, 86016, 516096, 110592, 221184, 294912, 221184, 147456, 36864, 6144, 24576, 24576, 49152, 98304, 122880, 172032, 147456, 49152, 98304, 122880, 172032, 147456, 1536, 7680, 4608, 9216, 53760, 46080, 61440, 0, 18432, 27648, 0, 36864, 82944, 18432, 9216, 55296, 268800, 161280, 376320, 322560, 0, 184320, 276480, 0, 368640, 0, 368640, 61440, 245760, 245760, 0, 18432, 147456, 73728, 55296, 36864, 55296, 193536, 165888, 110592, 221184, 110592, 221184, 294912, 221184, 147456, 663552, 165888, 746496, 331776, 331776, 18432, 147456, 73728, 55296, 36864, 82944, 18432, 9216, 55296, 276480, 276480, 110592, 497664, 331776, 165888, 995328, 829440, 1327104, 1327104, 1327104, 165888, 41472, 165888, 290304, 82944, 165888, 995328, 829440, 1327104, 1327104, 1327104, 580608, 2612736, 0, 870912, 0, 1161216, 663552, 165888, 746496, 331776, 331776, 1119744, 2612736, 1119744, 746496, 1492992, 2985984, 124416, 248832, 497664, 497664, 124416, 746496, 124416, 248832, 497664, 497664, 124416, 746496, 1990656, 4478976, 3483648, 2985984, 2985984, 1990656, 165888, 41472, 165888, 290304, 82944, 373248, 124416, 124416, 0, 497664, 41472, 62208, 6912, 13824, 13824, 41472, 110592, 27648, 55296, 13824, 41472, 110592, 27648, 55296, 165888, 41472, 165888, 290304, 82944, 110592, 110592, 0, 552960, 995328, 221184, 55296, 193536, 165888, 110592, 221184, 276480, 276480, 110592, 497664, 331776, 259200, 51840, 414720, 207360, 0, 129600, 259200, 1166400, 777600, 0, 0, 466560, 699840, 699840, 466560, 1866240, 0, 259200, 51840, 414720, 207360, 0, 51840, 129600, 233280, 51840, 0, 25920, 5184, 41472, 20736, 165888, 41472, 165888, 290304, 82944, 41472, 0, 145152, 62208, 124416, 165888, 995328, 829440, 1327104, 1327104, 1327104, 165888, 41472, 165888, 290304, 82944, 165888, 995328, 829440, 1327104, 1327104, 1327104, 580608, 2612736, 0, 870912, 0, 1161216, 663552, 165888, 746496, 331776, 331776, 165888, 41472, 165888, 290304, 82944, 145152, 580608, 725760, 145152, 725760, 290304, 373248, 124416, 124416, 0, 497664, 124416, 248832, 497664, 497664, 124416, 746496, 10368, 0, 31104, 62208, 82944, 93312, 31104, 31104, 0, 124416, 373248, 124416, 124416, 0, 497664, 663552, 165888, 746496, 331776, 331776, 13824, 41472, 110592, 27648, 55296, 165888, 41472, 165888, 290304, 82944, 110592, 110592, 0, 552960, 995328, 221184, 55296, 193536, 165888, 110592, 221184, 276480, 276480, 110592, 497664, 331776, 17280, 120960, 34560, 138240, 0, 120960, 241920, 0, 1088640, 725760, 0, 103680, 138240, 172800, 207360, 0, 138240, 414720, 1105920, 276480, 552960, 0, 165888, 41472, 165888, 290304, 82944, 145152, 580608, 725760, 145152, 725760, 290304, 373248, 124416, 124416, 0, 497664, 124416, 248832, 497664, 497664, 124416, 746496, 57600, 230400, 230400, 0, 0, 161280, 0, 120960, 80640, 0, 103680, 138240, 172800, 207360, 0, 184320, 276480, 0, 368640, 0, 414720, 622080, 69120, 138240, 0, 829440, 184320, 92160, 552960, 0, 184320, 276480, 0, 368640, 0, 414720, 622080, 69120, 138240, 0, 829440, 184320, 92160, 552960, 0, 16128, 96768, 16128, 32256, 129024, 870912, 580608, 677376, 580608, 774144, 16128, 96768, 16128, 32256, 129024, 96768, 64512, 64512, 161280, 193536, 129024, 258048, 1161216, 0, 258048, 516096, 12096, 24192, 0, 108864, 72576, 48384, 96768, 24192, 217728, 48384, 108864, 0, 870912, 870912, 653184, 435456, 508032, 145152, 362880, 508032, 435456, 387072, 258048, 322560, 64512, 129024, 193536, 387072, 145152, 387072, 193536, 96768, 64512, 64512, 161280, 193536, 10368, 0, 31104, 62208, 82944, 93312, 31104, 31104, 0, 124416, 373248, 124416, 124416, 0, 497664, 663552, 165888, 746496, 331776, 331776, 13824, 41472, 110592, 27648, 55296, 165888, 41472, 165888, 290304, 82944, 110592, 110592, 0, 552960, 995328, 221184, 55296, 193536, 165888, 110592, 221184, 276480, 276480, 110592, 497664, 331776, 17280, 120960, 34560, 138240, 0, 120960, 241920, 0, 1088640, 725760, 0, 103680, 138240, 172800, 207360, 0, 138240, 414720, 1105920, 276480, 552960, 0, 165888, 41472, 165888, 290304, 82944, 145152, 580608, 725760, 145152, 725760, 290304, 373248, 124416, 124416, 0, 497664, 124416, 248832, 497664, 497664, 124416, 746496, 192, 1728, 384, 1728, 12096, 3456, 13824, 1152, 3072, 1536, 1728, 12096, 3456, 13824, 12096, 24192, 0, 108864, 72576, 10368, 13824, 17280, 20736, 13824, 41472, 110592, 27648, 55296, 1152, 1152, 5760, 2304, 9216, 0, 21504, 6144, 1536, 7680, 4608, 9216, 1728, 12096, 3456, 13824, 12096, 24192, 0, 108864, 72576, 10368, 13824, 17280, 20736, 13824, 41472, 110592, 27648, 55296, 12096, 24192, 0, 108864, 72576, 48384, 96768, 24192, 217728, 48384, 108864, 0, 870912, 870912, 653184, 435456, 508032, 145152, 362880, 508032, 435456, 10368, 0, 31104, 62208, 82944, 13824, 41472, 110592, 27648, 55296, 17280, 120960, 34560, 138240, 0, 41472, 0, 145152, 62208, 124416, 13824, 41472, 110592, 27648, 55296, 165888, 41472, 165888, 290304, 82944, 110592, 110592, 0, 552960, 995328, 221184, 55296, 193536, 165888, 110592, 221184, 276480, 276480, 110592, 497664, 331776, 1152, 1152, 5760, 2304, 1152, 1152, 5760, 2304, 28800, 40320, 34560, 0, 4608, 6912, 0, 9216, 82944, 18432, 9216, 55296, 43008, 21504, 107520, 0, 86016, 36864, 6144, 24576, 24576, 1536, 7680, 4608, 9216, 53760, 46080, 61440, 0, 18432, 27648, 0, 36864, 82944, 18432, 9216, 55296, 1728, 12096, 3456, 13824, 12096, 24192, 0, 108864, 72576, 10368, 13824, 17280, 20736, 13824, 41472, 110592, 27648, 55296, 12096, 24192, 0, 108864, 72576, 48384, 96768, 24192, 217728, 48384, 108864, 0, 870912, 870912, 653184, 435456, 508032, 145152, 362880, 508032, 435456, 10368, 0, 31104, 62208, 82944, 13824, 41472, 110592, 27648, 55296, 17280, 120960, 34560, 138240, 0, 41472, 0, 145152, 62208, 124416, 13824, 41472, 110592, 27648, 55296, 165888, 41472, 165888, 290304, 82944, 110592, 110592, 0, 552960, 995328, 221184, 55296, 193536, 165888, 110592, 221184, 276480, 276480, 110592, 497664, 331776, 12096, 24192, 0, 108864, 72576, 48384, 96768, 24192, 217728, 48384, 108864, 0, 870912, 870912, 653184, 435456, 508032, 145152, 362880, 508032, 435456, 193536, 387072, 145152, 387072, 193536, 870912, 580608, 677376, 580608, 774144, 48384, 96768, 24192, 217728, 48384, 435456, 217728, 1524096, 1524096, 435456, 1741824, 193536, 387072, 145152, 387072, 193536, 108864, 0, 870912, 870912, 653184, 435456, 6967296, 6096384, 0, 7838208, 870912, 1741824, 6967296, 6096384, 0, 7838208, 870912, 1741824, 3919104, 3265920, 1959552, 653184, 5225472, 2612736, 1741824, 1306368, 2177280, 1741824, 2177280, 2612736, 2540160, 0, 4064256, 0, 1524096, 1016064, 145152, 580608, 725760, 145152, 725760, 290304, 1088640, 2177280, 725760, 2903040, 2903040, 0, 2540160, 0, 4064256, 0, 1524096, 1016064, 1741824, 1306368, 2177280, 1741824, 2177280, 2612736, 10368, 0, 31104, 62208, 82944, 93312, 31104, 31104, 0, 124416, 373248, 124416, 124416, 0, 497664, 663552, 165888, 746496, 331776, 331776, 13824, 41472, 110592, 27648, 55296, 165888, 41472, 165888, 290304, 82944, 110592, 110592, 0, 552960, 995328, 221184, 55296, 193536, 165888, 110592, 221184, 276480, 276480, 110592, 497664, 331776, 17280, 120960, 34560, 138240, 0, 120960, 241920, 0, 1088640, 725760, 0, 103680, 138240, 172800, 207360, 0, 138240, 414720, 1105920, 276480, 552960, 0, 165888, 41472, 165888, 290304, 82944, 145152, 580608, 725760, 145152, 725760, 290304, 373248, 124416, 124416, 0, 497664, 124416, 248832, 497664, 497664, 124416, 746496, 13824, 41472, 110592, 27648, 55296, 165888, 41472, 165888, 290304, 82944, 110592, 110592, 0, 552960, 995328, 221184, 55296, 193536, 165888, 110592, 221184, 276480, 276480, 110592, 497664, 331776, 165888, 995328, 829440, 1327104, 1327104, 1327104, 165888, 41472, 165888, 290304, 82944, 165888, 995328, 829440, 1327104, 1327104, 1327104, 580608, 2612736, 0, 870912, 0, 1161216, 663552, 165888, 746496, 331776, 331776, 110592, 110592, 0, 552960, 995328, 221184, 110592, 110592, 0, 552960, 995328, 221184, 2764800, 2764800, 1105920, 4976640, 3317760, 0, 8957952, 8957952, 4976640, 2985984, 1990656, 7962624, 442368, 442368, 221184, 221184, 1769472, 884736, 276480, 276480, 110592, 497664, 331776, 193536, 1741824, 580608, 967680, 580608, 1161216, 165888, 995328, 829440, 1327104, 1327104, 1327104, 110592, 110592, 0, 552960, 995328, 221184, 442368, 442368, 221184, 221184, 1769472, 884736, 552960, 1935360, 1658880, 1105920, 2211840, 0, 552960, 1935360, 1658880, 1105920, 2211840, 0, 110592, 110592, 0, 552960, 995328, 221184, 1990656, 4478976, 3483648, 2985984, 2985984, 1990656, 995328, 995328, 331776, 2322432, 2322432, 1990656, 1152, 1152, 5760, 2304, 1152, 1152, 5760, 2304, 28800, 40320, 34560, 0, 4608, 6912, 0, 9216, 1152, 1152, 5760, 2304, 1152, 1152, 5760, 2304, 28800, 40320, 34560, 0, 4608, 6912, 0, 9216, 57600, 230400, 230400, 0, 0, 161280, 0, 120960, 80640, 0, 103680, 138240, 172800, 207360, 0, 18432, 27648, 0, 36864, 41472, 62208, 6912, 13824, 82944, 18432, 9216, 55296, 663552, 165888, 746496, 331776, 331776, 18432, 147456, 73728, 55296, 36864, 82944, 18432, 9216, 55296, 276480, 276480, 110592, 497664, 331776, 172032, 129024, 0, 0, 344064, 43008, 21504, 107520, 0, 86016, 107520, 0, 752640, 537600, 215040, 0, 688128, 516096, 0, 86016, 516096, 110592, 221184, 294912, 221184, 147456, 36864, 6144, 24576, 24576, 49152, 98304, 122880, 172032, 147456, 49152, 98304, 122880, 172032, 147456, 1536, 7680, 4608, 9216, 53760, 46080, 61440, 0, 18432, 27648, 0, 36864, 82944, 18432, 9216, 55296, 268800, 161280, 376320, 322560, 0, 184320, 276480, 0, 368640, 0, 368640, 61440, 245760, 245760, 0, 18432, 147456, 73728, 55296, 36864, 55296, 193536, 165888, 110592, 221184, 110592, 221184, 294912, 221184, 147456, 663552, 165888, 746496, 331776, 331776, 18432, 147456, 73728, 55296, 36864, 82944, 18432, 9216, 55296, 276480, 276480, 110592, 497664, 331776, 1152, 1152, 5760, 2304, 1152, 1152, 5760, 2304, 28800, 40320, 34560, 0, 4608, 6912, 0, 9216, 1152, 1152, 5760, 2304, 1152, 1152, 5760, 2304, 28800, 40320, 34560, 0, 4608, 6912, 0, 9216, 57600, 230400, 230400, 0, 0, 161280, 0, 120960, 80640, 0, 103680, 138240, 172800, 207360, 0, 18432, 27648, 0, 36864, 41472, 62208, 6912, 13824, 82944, 18432, 9216, 55296, 1152, 1152, 5760, 2304, 1152, 1152, 5760, 2304, 28800, 40320, 34560, 0, 4608, 6912, 0, 9216, 1152, 1152, 5760, 2304, 1152, 1152, 5760, 2304, 28800, 40320, 34560, 0, 4608, 6912, 0, 9216, 57600, 230400, 230400, 0, 0, 161280, 0, 120960, 80640, 0, 103680, 138240, 172800, 207360, 0, 18432, 27648, 0, 36864, 41472, 62208, 6912, 13824, 82944, 18432, 9216, 55296, 288000, 403200, 345600, 0, 0, 460800, 691200, 0, 921600, 0, 0, 460800, 691200, 0, 921600, 0, 0, 161280, 967680, 161280, 322560, 1290240, 0, 120960, 241920, 0, 1088640, 725760, 0, 645120, 0, 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29859840, 44789760, 29859840, 67184640, 44789760, 0, 9953280, 9953280, 3317760, 23224320, 23224320, 19906560, 0, 9953280, 9953280, 3317760, 23224320, 23224320, 19906560, 0, 1327104, 3981312, 2654208, 9289728, 1327104, 0, 5308416, 11943936, 35831808, 31850496, 3981312, 11943936, 3981312, 7962624, 5308416, 15925248, 13271040, 10616832, 5308416, 0, 21233664, 83607552, 18579456, 74317824, 83607552, 65028096, 18579456, 74317824, 1327104, 3981312, 2654208, 9289728, 1327104, 0, 5308416, 26542080, 15925248, 0, 42467328, 21233664, 5308416, 31850496, 1327104, 3981312, 2654208, 9289728, 1327104, 0, 5308416, 11943936, 35831808, 31850496, 3981312, 11943936, 3981312, 7962624, 5308416, 15925248, 13271040, 10616832, 5308416, 0, 21233664, 83607552, 18579456, 74317824, 83607552, 65028096, 18579456, 74317824, 1327104, 3981312, 2654208, 9289728, 1327104, 0, 5308416, 26542080, 15925248, 0, 42467328, 21233664, 5308416, 31850496, 1327104, 3981312, 2654208, 9289728, 1327104, 0, 5308416, 11943936, 35831808, 31850496, 3981312, 11943936, 3981312, 7962624, 5308416, 15925248, 13271040, 10616832, 5308416, 0, 21233664, 83607552, 18579456, 74317824, 83607552, 65028096, 18579456, 74317824, 1327104, 3981312, 2654208, 9289728, 1327104, 0, 5308416, 26542080, 15925248, 0, 42467328, 21233664, 5308416, 31850496, 165888, 995328, 829440, 1327104, 1327104, 1327104, 8957952, 8957952, 4976640, 2985984, 1990656, 7962624, 6635520, 1658880, 7464960, 3317760, 3317760, 0, 1327104, 3981312, 2654208, 9289728, 1327104, 0, 5308416, 1327104, 3981312, 2654208, 9289728, 1327104, 0, 5308416, 1327104, 3981312, 2654208, 9289728, 1327104, 0, 5308416, 165888, 995328, 829440, 1327104, 1327104, 1327104, 165888, 41472, 165888, 290304, 82944, 165888, 995328, 829440, 1327104, 1327104, 1327104, 580608, 2612736, 0, 870912, 0, 1161216, 663552, 165888, 746496, 331776, 331776, 165888, 995328, 829440, 1327104, 1327104, 1327104, 8957952, 8957952, 4976640, 2985984, 1990656, 7962624, 6635520, 1658880, 7464960, 3317760, 3317760, 0, 1327104, 3981312, 2654208, 9289728, 1327104, 0, 5308416, 1327104, 3981312, 2654208, 9289728, 1327104, 0, 5308416, 1327104, 3981312, 2654208, 9289728, 1327104, 0, 5308416, 2903040, 4644864, 0, 3483648, 0, 4644864, 5225472, 15676416, 2612736, 5225472, 18289152, 7838208, 15676416, 6967296, 6096384, 0, 7838208, 870912, 1741824, 1161216, 1161216, 6967296, 1161216, 2322432, 1161216, 6967296, 3981312, 3981312, 1990656, 3317760, 3317760, 1327104, 165888, 995328, 829440, 1327104, 1327104, 1327104, 5225472, 2985984, 4478976, 2985984, 6718464, 4478976, 995328, 995328, 331776, 2322432, 2322432, 1990656, 995328, 995328, 331776, 2322432, 2322432, 1990656, 165888, 995328, 829440, 1327104, 1327104, 1327104, 8957952, 8957952, 4976640, 2985984, 1990656, 7962624, 6635520, 1658880, 7464960, 3317760, 3317760, 0, 1327104, 3981312, 2654208, 9289728, 1327104, 0, 5308416, 1327104, 3981312, 2654208, 9289728, 1327104, 0, 5308416, 1327104, 3981312, 2654208, 9289728, 1327104, 0, 5308416, 71663616, 80621568, 44789760, 62705664, 80621568, 44789760, 17915904, 71663616, 80621568, 44789760, 62705664, 80621568, 44789760, 17915904, 19906560, 44789760, 34836480, 29859840, 29859840, 19906560, 0, 5971968, 26873856, 23887872, 14929920, 26873856, 23887872, 11943936, 1990656, 17915904, 17915904, 0, 11943936, 9953280, 11943936, 55738368, 71663616, 47775744, 15925248, 47775744, 15925248, 31850496, 39813120, 39813120, 19906560, 33177600, 33177600, 13271040, 0, 1658880, 9953280, 8294400, 13271040, 13271040, 13271040, 0, 52254720, 29859840, 44789760, 29859840, 67184640, 44789760, 0, 9953280, 9953280, 3317760, 23224320, 23224320, 19906560, 0, 9953280, 9953280, 3317760, 23224320, 23224320, 19906560, 0, 1327104, 3981312, 2654208, 9289728, 1327104, 0, 5308416, 11943936, 35831808, 31850496, 3981312, 11943936, 3981312, 7962624, 5308416, 15925248, 13271040, 10616832, 5308416, 0, 21233664, 83607552, 18579456, 74317824, 83607552, 65028096, 18579456, 74317824, 1327104, 3981312, 2654208, 9289728, 1327104, 0, 5308416, 26542080, 15925248, 0, 42467328, 21233664, 5308416, 31850496, 1327104, 3981312, 2654208, 9289728, 1327104, 0, 5308416, 11943936, 35831808, 31850496, 3981312, 11943936, 3981312, 7962624, 5308416, 15925248, 13271040, 10616832, 5308416, 0, 21233664, 83607552, 18579456, 74317824, 83607552, 65028096, 18579456, 74317824, 1327104, 3981312, 2654208, 9289728, 1327104, 0, 5308416, 26542080, 15925248, 0, 42467328, 21233664, 5308416, 31850496, 1327104, 3981312, 2654208, 9289728, 1327104, 0, 5308416, 11943936, 35831808, 31850496, 3981312, 11943936, 3981312, 7962624, 5308416, 15925248, 13271040, 10616832, 5308416, 0, 21233664, 83607552, 18579456, 74317824, 83607552, 65028096, 18579456, 74317824, 1327104, 3981312, 2654208, 9289728, 1327104, 0, 5308416, 26542080, 15925248, 0, 42467328, 21233664, 5308416, 31850496, 5806080, 26127360, 0, 8709120, 0, 11612160, 0, 18579456, 27869184, 18579456, 18579456, 37158912, 27869184, 18579456, 10450944, 13934592, 27869184, 10450944, 20901888, 13934592, 27869184, 18579456, 27869184, 18579456, 18579456, 37158912, 27869184, 18579456, 26127360, 10450944, 10450944, 26127360, 20901888, 36578304, 10450944, 15676416, 78382080, 94058496, 109734912, 94058496, 62705664, 15676416, 94058496, 5225472, 15676416, 2612736, 5225472, 18289152, 7838208, 15676416, 26127360, 10450944, 10450944, 26127360, 20901888, 36578304, 10450944, 18289152, 146313216, 36578304, 146313216, 164602368, 18289152, 91445760, 36578304, 54867456, 62705664, 23514624, 62705664, 15676416, 0, 62705664, 15676416, 78382080, 94058496, 109734912, 94058496, 62705664, 15676416, 94058496, 41803776, 62705664, 41803776, 48771072, 13934592, 62705664, 41803776, 36578304, 0, 54867456, 36578304, 18289152, 48771072, 24385536, 54867456, 62705664, 23514624, 62705664, 15676416, 0, 62705664, 6967296, 6096384, 0, 7838208, 870912, 1741824, 1741824, 12192768, 6967296, 1741824, 13934592, 3483648, 6967296, 1161216, 1161216, 6967296, 1161216, 2322432, 1161216, 6967296, 1161216, 1161216, 6967296, 1161216, 2322432, 1161216, 6967296, 41803776, 62705664, 41803776, 48771072, 13934592, 62705664, 41803776, 1161216, 1161216, 6967296, 1161216, 2322432, 1161216, 6967296, 4644864, 6967296, 4644864, 4644864, 9289728, 6967296, 4644864, 1161216, 1161216, 6967296, 1161216, 2322432, 1161216, 6967296, 41803776, 62705664, 41803776, 48771072, 13934592, 62705664, 41803776, 11943936, 35831808, 31850496, 3981312, 11943936, 3981312, 7962624, 11943936, 35831808, 31850496, 3981312, 11943936, 3981312, 7962624, 1990656, 17915904, 17915904, 0, 11943936, 9953280, 11943936, 9953280, 9953280, 3317760, 23224320, 23224320, 19906560, 0, 9953280, 9953280, 3317760, 23224320, 23224320, 19906560, 0, 1327104, 3981312, 2654208, 9289728, 1327104, 0, 5308416, 165888, 995328, 829440, 1327104, 1327104, 1327104, 8957952, 8957952, 4976640, 2985984, 1990656, 7962624, 6635520, 1658880, 7464960, 3317760, 3317760, 0, 1327104, 3981312, 2654208, 9289728, 1327104, 0, 5308416, 1327104, 3981312, 2654208, 9289728, 1327104, 0, 5308416, 1327104, 3981312, 2654208, 9289728, 1327104, 0, 5308416, 26127360, 10450944, 10450944, 26127360, 20901888, 36578304, 10450944, 5971968, 26873856, 23887872, 14929920, 26873856, 23887872, 11943936, 17915904, 17915904, 31352832, 35831808, 40310784, 31352832, 26873856, 5971968, 26873856, 23887872, 14929920, 26873856, 23887872, 11943936, 40310784, 47029248, 6718464, 53747712, 26873856, 40310784, 26873856, 17915904, 17915904, 31352832, 35831808, 40310784, 31352832, 26873856, 8957952, 8957952, 4976640, 2985984, 1990656, 7962624, 8957952, 8957952, 4976640, 2985984, 1990656, 7962624, 995328, 995328, 331776, 2322432, 2322432, 1990656, 4644864, 6967296, 4644864, 4644864, 9289728, 6967296, 4644864, 4644864, 6967296, 4644864, 4644864, 9289728, 6967296, 4644864, 1990656, 17915904, 17915904, 0, 11943936, 9953280, 11943936, 8957952, 8957952, 4976640, 2985984, 1990656, 7962624, 8957952, 8957952, 4976640, 2985984, 1990656, 7962624, 995328, 995328, 331776, 2322432, 2322432, 1990656, 4644864, 6967296, 4644864, 4644864, 9289728, 6967296, 4644864, 4644864, 6967296, 4644864, 4644864, 9289728, 6967296, 4644864, 1990656, 17915904, 17915904, 0, 11943936, 9953280, 11943936, 1119744, 1119744, 1119744, 10077696, 7838208, 4478976, 4478976, 1119744, 1119744, 1119744, 10077696, 7838208, 4478976, 4478976, 1119744, 1119744, 1119744, 10077696, 7838208, 4478976, 4478976, 10077696, 0, 0, 70543872, 70543872, 60466176, 90699264, 60466176, 54867456, 62705664, 23514624, 62705664, 15676416, 0, 62705664, 17915904, 17915904, 31352832, 35831808, 40310784, 31352832, 26873856, 17915904, 17915904, 31352832, 35831808, 40310784, 31352832, 26873856, 26127360, 10450944, 10450944, 26127360, 20901888, 36578304, 10450944, 15676416, 78382080, 94058496, 109734912, 94058496, 62705664, 15676416, 94058496, 5225472, 15676416, 2612736, 5225472, 18289152, 7838208, 15676416, 26127360, 10450944, 10450944, 26127360, 20901888, 36578304, 10450944, 18289152, 146313216, 36578304, 146313216, 164602368, 18289152, 91445760, 36578304, 54867456, 62705664, 23514624, 62705664, 15676416, 0, 62705664, 15676416, 78382080, 94058496, 109734912, 94058496, 62705664, 15676416, 94058496, 1119744, 1119744, 1119744, 10077696, 7838208, 4478976, 4478976, 1119744, 1119744, 1119744, 10077696, 7838208, 4478976, 4478976, 1119744, 1119744, 1119744, 10077696, 7838208, 4478976, 4478976, 10077696, 0, 0, 70543872, 70543872, 60466176, 90699264, 60466176, 54867456, 62705664, 23514624, 62705664, 15676416, 0, 62705664, 17915904, 17915904, 31352832, 35831808, 40310784, 31352832, 26873856, 17915904, 17915904, 31352832, 35831808, 40310784, 31352832, 26873856, 26127360, 10450944, 10450944, 26127360, 20901888, 36578304, 10450944, 5971968, 26873856, 23887872, 14929920, 26873856, 23887872, 11943936, 17915904, 17915904, 31352832, 35831808, 40310784, 31352832, 26873856, 5971968, 26873856, 23887872, 14929920, 26873856, 23887872, 11943936, 40310784, 47029248, 6718464, 53747712, 26873856, 40310784, 26873856, 17915904, 17915904, 31352832, 35831808, 40310784, 31352832, 26873856, 1492992, 5971968, 13436928, 2985984, 13436928, 13436928, 2985984, 29859840, 53747712, 41803776, 5971968, 53747712, 35831808, 47775744, 13436928, 40310784, 53747712, 40310784, 80621568, 120932352, 26873856, 107495424, 5971968, 26873856, 23887872, 14929920, 26873856, 23887872, 11943936, 13436928, 40310784, 53747712, 40310784, 80621568, 120932352, 26873856, 107495424, 13436928, 40310784, 53747712, 40310784, 80621568, 120932352, 26873856, 107495424, 5971968, 26873856, 23887872, 14929920, 26873856, 23887872, 11943936, 29859840, 53747712, 41803776, 5971968, 53747712, 35831808, 47775744, 53747712, 161243136, 214990848, 188116992, 80621568, 214990848, 134369280, 161243136, 47775744, 71663616, 191102976, 191102976, 167215104, 191102976, 167215104, 47775744, 14929920, 59719680, 134369280, 29859840, 134369280, 134369280, 29859840, 0, 53747712, 161243136, 214990848, 188116992, 80621568, 214990848, 134369280, 161243136, 47775744, 71663616, 191102976, 191102976, 167215104, 191102976, 167215104, 47775744, 11943936, 11943936, 107495424, 47775744, 35831808, 107495424, 35831808, 71663616, 124416, 248832, 497664, 497664, 124416, 746496, 497664, 995328, 1990656, 1990656, 746496, 497664, 1990656, 4478976, 3483648, 2985984, 2985984, 1990656, 1990656, 4478976, 3483648, 2985984, 2985984, 1990656, 124416, 248832, 497664, 497664, 124416, 746496, 5225472, 2985984, 4478976, 2985984, 6718464, 4478976, ...

________________________

Merci beaucoup Jean-Marc !

(This seq will enter soon the OEIS, I hope ).

________________________

Feb 2, 2024 update

Two new "boomerang sequences" A and B

A: « Replace each digit d of S by its 6th power: the sequence S remains identical to itself if we follow each result with a comma. »

B: « Replace each digit d of S by its 8th power: the sequence S remains identical to itself if we follow each result with a comma. »

A = 0, 1, 4096, 0, 531441, 46656, 0, 15625, 729, 1, 4096, 4096, 1,

dig 0 1 4 0 9 6 0 5 3 1 4 4 1

4096, 46656, 46656, 15625, 46656, 0, 1,15625, 46656, 64, 1,15625,

4 6 6 5 6 0 1 5 6 2 5

117649, 64, 531441, 1, 4096, 0, 531441, 46656, 4096, 0, 531441, 46656

7 2 9 1 4 0 9 6 4 0 9 6

B = 0, 1, 256, 390625, 1679616, 6561, 43046721, 0, 1679616, 256, 390625

dig 0 1 2 5 6 3 9 0 6 2 5

1, 1679616, 5764801, 43046721, 1679616, 1, 1679616, 1679616, 390625, 1679616, 1,

1 6 7 9 6 1 6 6 5 6 1

These seqs have just been submitted to the OEIS, here (A) and there (B).

_________________________________________________

(Boomerangs updates so far)

BOOMERANG #1

https://cinquantesignes.blogspot.com/2024/01/the-same-sequence-but-differently.html

BOOMERANG #2

https://cinquantesignes.blogspot.com/2024/02/boomerang-sequences.html

BOOMERANG #3

https://cinquantesignes.blogspot.com/2024/02/damn-dam-madness.html

BOOMERANG #4

https://cinquantesignes.blogspot.com/2024/02/a-last-boomerang.html

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