Let's transform a sequence of integers into another one, where only prime terms are left.
This is done by expelling a digit from a(n) and sending it towards a(n+1); there, this "bullet" digit will replace (and expel in turn) a digit of a(n+2) towards a(n+3), and so on. After this "chain reaction" is completed, all "hit" terms will be prime.
As usual, we want the sequence S to be the lexicographically earliest one made of distinct integers > 0 with this property. If I'm not wrong (caveat!) S starts like this:
S = 1, 10, 101, 103, 107, 109, 111, 11, 12, 2, 3, 4, 13, 14, 17, 15, 5, 6, 21, 7, 16, 27, 8, 19, 18, ...
Explanation:
In yellow are the successive expelled digits – and we see after S the new sequence P of prime terms only:
S = 1, 10, 101, 103, 107, 109, 111, 11, 12, 2, 3, 4, 13, 14, 17, 15, 5, 6, 21, 7, 16, 27, 8, 19, 18, ...
P = . 11, 101, 103, 107, 109, 101, 11, 11, 2, 2, 3, 43, 11, 47, 11, 5, 5, 61, 2, 17, 67, 2, 89, 11, ...
S is not in the OEIS.
February 23rd update:
A variant with odd terms (instead of primes):
> Let's transform a sequence of distinct integers >0 into another one, where only odd terms are left.
T = 1,2,11,3,4,13,5,6,15,7,8,17,9,10,101,103,105,107,109,111,12,19,14,21,23,25,27,29,31,16,33,18,35,20,113,22,37,24,39,26,41,43,45,47,49,51,28,53,30,115,32,55,34,57,36,59,38,61,63,65,67,69,71,40,117,42,73,44,75,46,77,48,79,50,119,52,81,83,85,87,89,91,54,93,56,95,58,97,60,121,99,62,...
(the blue digits are the expelled ones).
Coming soon (?) the "only even terms are left".
February 24th update:
A variant with even terms (instead of odd):
> Let's transform a sequence of distinct integers >0 into another one, where only even terms are left.
U = 1, 10, 12, 14, 16, 18, 20, 3, 22, 2, 4, 5, 24, 6, 7, 26, 8, 9, 28, 11, 30, 32, 34, 36, 38, 40, 13, 42, 15, 44, 17, 46, 19, 48, 21, 50, 52, 54, 56, 58, 60, 23, 62, 25, 64, 27, 66, 29, 68, 31, ...
February 25th update:
A variant with Fibonacci terms (instead of even):
> Let's transform a sequence of distinct integers >=0 into another one, where only Fibonacci terms are left.
V = 0, 1, 2, 3, 4, 30, 610, 611, 5, 6, 110, 7, 307, 612, 8, 9, 80, 613, 10, 614, 31, 13, 20, 615, 15, 21, 22, 11, 23, 41, 32, 51, 25, 61, 210, 71, 317, 24, 33, 14, 26, 310, 34, 12, 81, 19, ...
February 26th update:
A variant with square terms (instead of Fibonacci):
> Let's transform a sequence of distinct integers >0 into another one, where only square terms are left.
W = 1, 2, 15, 3, 16, 4, 5, 20, 100, 101, 6, 10, 102, 25, 35, 26, 45, 7, 506, 103, 36, 46, 8, 11, 9, 40, 104, 19, 56, 21, 66, 12, 55, 22, 65, 13, 76, 516, 80, 105, 23, 86, 31, 96, 41, 81, 82, 75, 526, 85, 51, 24, 29, 95, 42, 111, 83, 124, 84, 39, 224, 121, 131, 324, 424, 49, 59, 27, 536, 524, 28, 61, 14, ...
February 27th update:
All those seqs have now been computed (and corrected!) by Carole D. – many thanks Carole!
They are in the OEIS (or will be soon):
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