Replace my twins, triplets, etc. by 1
(English version by Google below)
French
On
choisit (ici) n = 1, lequel on multiplie par 2, puis le résultat par 2, puis le
résultat par 2, etc.
Dès que
l’une des itérations contient un bloc de deux ou plusieurs chiffres adjacents identiques, ce bloc
est remplacé par le chiffre 1. Et on itère à nouveau.
Pour
a(1) = 1 la suite F entre dans une boucle de 42 termes :
F = 1, 2, 4,
8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 6136, 12272, 1172, 172, 344, 31, 62, 124,
248, 496, 992, 12, 24, 48,
96, 192, 384, 768, 1536, 3072, 6144, 611, 61, 122, 11, 1, 2, 4, 8,
16,...
Y a-t-il
des boucles plus courtes ? Plus longues ? Certaines valeurs de a(n) n’entrent-elles
jamais dans une boucle ?
Une suite intéressante S qui trouverait peut-être sa place dans l’OEIS peut se développer à l’infini : on remplace le premier terme de la boucle à venir par le plus petit entier absent jusque là. L’itération se poursuivrait (avec 3 ici), s’interromperait avec un terme déjà présent, reprendrait avec un terme « frais », etc.
S = 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 6136, 12272, 1172, 172, 344, 31, 62, 124, 248, 496, 992, 12, 24, 48, 96, 192, 384, 768, 1536, 3072, 6144, 611, 61, 122, 11, 3, 6, 5, ...
English
We
choose (here) n = 1, which we multiply by 2, then the result by 2, then the
result by 2, etc.
As soon
as one of the iterations contains a block of two or more identical adjacent digits, this block is
replaced by the digit 1. And we iterate again.
For a(1)
= 1 the sequence F enters a loop of 42 terms:
F = 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 6136, 12272, 1172, 172, 344, 31, 62, 124, 248, 496, 992, 12, 24, 48, 96, 192, 384, 768, 1536, 3072, 6144, 611, 61, 122, 11, 1, 2, 4, 8, 16,...
Are
there shorter loops? Longer ? Do some values of a(n) never enter a loop?
An interesting sequence S which would perhaps find its place in the OEIS can be developed infinitely: we replace the first term of the loop to come by the smallest integer absent until then. The iteration would continue (with 3 here), interrupt with a term already present, resume with a “fresh” term, etc.
S = 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 6136, 12272, 1172, 172, 344, 31, 62, 124, 248, 496, 992, 12, 24, 48, 96, 192, 384, 768, 1536, 3072, 6144, 611, 61, 122, 11, 3, 6, 5, ...
____________________
Giorgos K. was quick to send this:
> Here are some remarks about your new seq:
Conjecture:
All the loops for different a(1) have length 42 or 8 (most of the times 42).
If s is the starting point of the loop, then here are the first loops with length NOT equal to 42:
a(1) s length
234 1872 8
371 371 8
468 1872 8
742 742 8
873 1872 8
925 371 8
936 1872 8
947 371 8
As we can see all the lengths not equal to 42 are equal to 8. This holds for the first million values of a(1): lengths are equal to 42 or 8.
Now about the second seq, here are the first 100 terms + plot and Log-plot for the first 10000 terms:
1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 6136, 12272, 1172, 172, 344, 31, 62, 124, 248, 496, 992, 12, 24, 48, 96, 192, 384, 768, 1536, 3072, 6144, 611, 61, 122, 11, 3, 6, 5, 10, 20, 40, 80, 160, 320, 640, 1280, 2560, 5120, 10240, 20480, 40960, 81920, 163840, 327680, 655360, 61360, 122720, 11720, 1720, 3440, 310, 620, 1240, 2480, 4960, 9920, 120, 240, 480, 960, 1920, 3840, 7680, 15360, 30720, 61440, 6110, 610, 1220, 110, 7, 14, 28, 56, 112, 9, 18, 36, 72, 144, 13, 26, 52,...
____________________
Next day update (Jan. 6th, 2024)
Hans Havermann:
> Giorgos: "All the loops for different a(1) have length 42 or 8."
The loops come in pairs. Known so far, minimums = 1 & 10 (lengths 42); minimums = 371 & 3710 (lengths 8); minimums = 370371 & 3703710 (lengths 8).
[1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 6136, 12272, 1172, 172, 344, 31, 62, 124, 248, 496, 992, 12, 24, 48, 96, 192, 384, 768, 1536, 3072, 6144, 611, 61, 122, 11, 1]
[371, 742, 1484, 2968, 5936, 11872, 1872, 3744, 371]
[370371, 740742, 1481484, 2962968, 5925936, 11851872, 1851872, 3703744, 370371]
[HH, some time later]:
I should have been able to figure this out sooner. The 371 (3710) & 370371 (3703710) extend to length-8 loops 370370371 (3703703710), 370370370371 (3703703703710), etc.
[370370371, 740740742, 1481481484, 2962962968, 5925925936, 11851851872, 1851851872, 3703703744, 370370371]
[370370370371, 740740740742, 1481481481484, 2962962962968, 5925925925936, 11851851851872, 1851851851872, 3703703703744, 370370370371]
Eric A:
Thanks for those nice pairs, Hans!
____________________
Next update (Jan. 7th, 2024)
– a private conversation with Giorgos:
EA
Hi Giorgos,
I like a lot this idea of replacing a chunk of adjacent equal digits by 1, then multiply the result by 2 and iterate.
We see that we can easily play with a few parameters to produce and study more variants.
1) replace the chunk by 2 (instead of 1), or 3, or 4, or ...42, or anything else;
2) replace the chunk by [the chunk + 1], or [the chunk + 2] or ... [the chunk + 42];
3) multiply the result by 3 instead of 2, or by 4, or by 5, etc.
The idea is to check if the new iterations behave like the first you spotted (loops of 8 and 42 — coming by pairs as Hans H. has found), etc.
Here is my first try by hand — ending indeed in a 16-loop (same rules as the first ones, but multiplication by 3 instead of 2):
T = 1,3,9,27,81,243,729,2187,6561,19683,59049,177147,11147,147,441,11,(1)
Second try, a(1)=2 and multiplication by 3 again:
U = 2, 6, 18, 54, 162, 4861458, 4374, 13122, 1311, 131, 393, 1179, 179, 537, 1611, 161, 483, 1449, 119, 19, 57, 171, 513, 1539, 4617, 13851, 41553, 4113, 413, 1239, 3717, 11151, 151, 453, 1359, 4077, 401, 1203, 3609, 10827, 32481, 97443, 9713, 29139, 87417, 262251, 26151, 78453, 235359, 706077, 70601, 211803, 21803, 65409, 196227, 19617, 58851, 5151, 15453, 46359, 139077, 13901, 41703, 125109, 375327, 1125981, 125981, 377943, 31943, 95829, 287487, 862461, 2587383, 7762149, 162149, 486447, 48617, 145851, 437553, 43713, 131139, 13139, 39417, 118251, 18251, 54753, 164259, 492777, 4981, 14763, 44289, 1289, 3867, 11601, 1601, 4803, 14409, 1109, 109, 327, 981, 2943, 8829, 129, 387, 1161, 161, 48, 1449, 119, 19, 57, 171, 513, 1539, 4617, 13852, 41553, 4113, 413, 1239, 3717, 11151, 151, 453, 1359, 4077, 401, 1203, 3609, 10827, 32481, 97443, 9713, 29139, 87417, 262251, 26151, 78453, 235359, 706077, 70601, 211803, 21803, 65409, 196227, 19617, 58851, 5151, 15453, 46359, 139077, 13901, 41703, 125109, 37532, 1125981, 125891, 377943, 3193, 95829, 287487, 862461, 2587383, 7762149, 162149, 486447, 48617, etc.
— I am lost and probably on a wrong track! What do you think?
GK
Let's start from the end...
Your sequence U should have stopped at the second 161...
U = 2, 6, 18, 54, 162, 486, 1458, 4374, 13122, 1311, 131, 393, 1179, 179, 537, 1611, 161, 483, 1449, 119, 19, 57, 171, 513, 1539, 4617, 13851, 41553, 4113, 413, 1239, 3717, 11151, 151, 453, 1359, 4077, 401, 1203, 3609, 10827, 32481, 97443, 9713, 29139, 87417, 262251, 26151, 78453, 235359, 706077, 70601, 211803, 21803, 65409, 196227, 19617, 58851, 5151, 15453, 46359, 139077, 13901, 41703, 125109, 375327, 1125981, 125981, 377943, 31943, 95829, 287487, 862461, 2587383, 7762149, 162149, 486447, 48617, 145851, 437553, 43713, 131139, 13139, 39417, 118251, 18251, 54753, 164259, 492777, 4921, 14763, 44289, 1289, 3867, 11601, 1601, 4803, 14409, 1109, 109, 327, 981, 2943, 8829, 129, 387, 1161, (161)
So, you have a loop of length 91.
3) Now, here are the lengths of loops of the first 100 a(1) for different multipliers:
2-> (see seq F opening this page){42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42,42}
3-> (see seqs T and U above){16,91,16,91,17,91,91,31,16,16,16,91,91,91,17,16,91,91,91,91,91,16,91,31,17,91,16,16,31,16,31,91,16,91,17,91,16,91,91,91,31,91,91,16,17,91,16,16,16,17,91,91,31,91,16,16,91,91,16,91,91,91,91,91,91,16,91,91,91,91,31,31,31,16,17,91,16,91,31,31,16,91,16,16,91,91,31,16,91,16,91,91,31,91,16,91,31,91,16,16}
4->{41,41,41,41,41,41,41,41,41,41,41,41,41,3,41,41,41,41,3,41,41,41,41,41,41,3,41,41,41,41,41,41,41,41,3,41,41,41,41,41,41,41,41,41,41,41,41,41,41,41,41,41,41,41,41,3,41,41,41,41,41,41,41,41,3,41,41,41,3,41,41,41,41,41,41,3,41,41,3,41,41,41,41,41,41,41,41,41,41,41,41,41,41,41,41,41,41,3,41,41}
5->{867,867,401,867,867,401,401,867,401,867,867,867,867,401,401,13,867,401,401,867,867,867,401,867,867,867,401,867,867,401,867,39,867,867,401,13,867,401,401,867,13,867,401,867,401,401,401,867,401,867,867,867,867,401,867,867,867,867,401,867,867,867,401,867,867,867,401,39,867,401,867,867,867,867,401,13,867,401,401,13,39,13,401,867,867,401,401,867,401,401,13,867,867,401,401,13,867,401,867,867}
6->{147,6,147,147,147,147,6,147,147,147,147,6,147,7,147,147,28,147,7,6,147,147,7,147,7,147,85,147,147,147,147,28,147,7,147,147,147,147,7,147,147,6,147,147,85,147,28,147,147,147,147,28,147,147,147,147,28,7,147,147,147,28,147,7,7,147,28,57,147,6,147,6,147,147,147,7,147,147,147,147,147,6,147,7,147,147,147,147,147,147,7,6,147,147,28,147,6,147,147,147}
7->{39,513,513,39,71,6,39,513,39,39,39,39,6,513,6,39,39,39,39,513,513,39,39,513,6,513,39,39,513,513,39,513,39,39,71,513,39,513,39,39,39,6,39,39,6,39,39,39,39,71,39,513,513,513,39,513,39,39,39,6,39,39,39,39,39,39,513,39,59,39,513,513,39,39,6,513,39,39,6,513,59,39,59,39,39,513,39,39,513,39,6,39,39,513,6,513,513,513,39,39}
8->{18,18,18,18,18,18,18,18,90,18,18,18,18,18,18,18,90,18,18,18,90,18,18,18,90,18,18,18,90,18,18,18,18,90,18,90,18,18,90,18,18,18,18,18,90,18,18,18,18,18,18,18,18,18,18,18,90,18,90,18,18,18,90,18,18,18,90,18,18,18,18,90,18,18,18,18,18,18,90,18,18,18,18,90,18,18,90,18,18,90,18,18,18,18,18,18,18,18,18,18}
9->{119,119,13,119,29,119,119,119,119,119,119,13,119,119,155,119,119,119,119,119,119,119,119,119,155,119,13,119,119,13,119,119,119,119,155,119,119,119,119,119,13,119,119,119,29,119,119,119,119,29,119,119,13,119,119,119,119,119,119,119,119,119,119,119,155,119,119,13,119,119,119,119,119,119,194,119,119,119,119,119,119,13,119,119,155,119,119,119,119,119,119,13,119,119,119,119,119,13,119,119}
10->{4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
42->{6348,6348,6348,6348,6348,6348,6348,6348,6348,6348,6348,6348,6348,6348,6348,6348,6348,6348,1263,6348,6348,6348,1263,6348,6348,6348,6348,6348,6348,6348}
2) If we substitute the chunk with [chunk+1] the sequences seem to explode to infinity...
V = 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 65636, 131272, 262544, 262545, 525090, 1050180, 2100360, 211360, 212360, 424720, 849440, 849450, 1698900, 169891, 339782, 349782, 699564, 6100564, 611564, 612564, 1225128, 1235128, 2470256, 4940512, 9881024, 9891024, 19782048, 39564096, 79128192, 158256384, 316512768, 633025536, 634025636, 1268051272, 2536102544, 2536102545, 5072205090, 5072305090, 10144610180, 10145610180, 20291220360, 20291230360, 40582460720, 81164921440, 81264921450, 162529842900, 16252984291, 32505968582...
1) Finally, if we replace the chunks with 2,3,4... instead of 1, we get loops of lengths:
—chunks replaced by 2, multiplication by 2:
{12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12} probably always 12
—chunks replaced by 2, multiplication by 3:{120,141,120,141,3,141,116,141,120,120,141,141,120,141,3,141,141,141,116,141,116,141,79,141,3,141,120,141,116,120,141,141,141,141,6,141,141,141,120,141,79,141,116,141,3,141,79,141,116,3,141,141,141,141,141,141,116,141,141,141,79,141,116,141,141,141,120,141,79,116,141,141,141,141,3,141,141,141,141,141,120,26,116,141,141,141,116,141,116,120,79,141,141,141,26,141,141,141,141,141}
etc.
—chunks replaced by 3, multiplication by 2:{14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14}
—chunks replaced by 3, multiplication by 3:{23,23,23,23,9,23,49,23,23,23,23,23,23,23,9,23,23,23,49,23,49,23,23,23,26,23,23,23,23,23,23,23,23,23,26,23,23,23,23,23,23,23,23,23,9,23,23,23,23,9,23,23,23,23,23,49,49,23,23,23,23,23,49,23,23,23,23,23,23,49,23,23,23,23,26,23,23,23,23,23,23,49,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23}
etc.
____________________
EA
Nice job, Giorgos – and fascinating results, for me: the regularity of the loops, even changing the initial conditions, is a question mark... It seems difficult to find rules producing a chaotic seq not entering a loop and not exploding too fast!
Merci, merci, merci Giorgos – the math universe is infinite and beautiful – wake up the babies!-)
____________________
Evening update:
HH
V = 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 65636, 131272, 262544, 262545, 525090, 1050180, 2100360, 211360, 212360, 424720, 849440, 849450, 1698900, 169891, 339782, 349782, 699564, 6100564, 611564, 612564, 1225128, 1235128, 2470256, 4940512, 9881024, 9891024, 19782048, 39564096, 79128192, 158256384, 316512768, 633025536, 634025636, 1268051272, 2536102544, 2536102545, 5072205090, 5072305090, 10144610180, 10145610180, 20291220360, 20291230360, 40582460720, 81164921440, 81264921450, 162529842900, 16252984291, 32505968582, ...
I like this one. And yes, while the numbers get quickly large, their eventual sizes seem to be largely confined, say (roughly) 10^15 to 10^55.
____________________
Many thanks, Hans – your definition of the confinement made my evening, I can't even read the last figures !-)
____________________
HH
Perhaps the attached graph explains it better:
EA
Waooooow, what a marvel! – and yet another mystery (the confinement)..._____________________
Midnight update
HH
I just calculated 100 million terms of V.
A graph of the final one million terms is attached. The smallest number in this region is 40584253184307148291. The largest is 165943212508653801005149589895625087149571210050257450. The chances of running into a duplicate of integers this large is very, very small. However, eventually it will happen (there are only a finite number of even very large integers) and when it does, we will have a loop.
What can I say more?
My thanks for the very clear explanations — and my gratitude for the wonderful blue graph!
____________________
Next day update by Giorgos K.
GK
Very nice observation by Hans!
Let's consider these numbers "random".
If you think about it, you have increasing and decreasing number of digits in the following situations:
a) Increasing
Because the number is doubled: That is everytime the starting digit is 5,6,7,8,9, When we have consecutive doublings this happens around 30% of the time.
Also when we have 99->100 we have +1 digit but in the next step we have 00->1 and -1 digit (so this doesn't count)
b) Decreasing
When we have 2 or more consecutive zeros
The probability of getting consecutive zeros depends on the length of the number.
If the number is small then the probability is small and "doubling" is winning.
If the number is very large then we have the opposite.
So, I computed (an approximation) of the "sweet spot" : That is "how big must the integer be in order for it to have consecutive zeros with probability 30%"
It turns out that this number is (approximately) 10^40. This means that above this number the zeros are winning and the digits decrease and vice versa. So, this system "fights" between these two states and we get the "zone of equilibrium". I agree with Hans that the system must reach a loop eventually.
____________________
Mid-March (very late) update
The wonderful February-loop update by Hans is there, on his blog (which I've read only today), and hereunder.
____________________
HH
> Confined (a loop)
> I found a loop in Éric Angelini's "confined" sequence (about which I wrote last month). Term #60614674264 (= 27651356989742597468495745) is a duplicate of term #18563532230. Differences in the lead-up terms are highlighted here:
#18563532226 6912789247435649367123936 #60614674260 6912789247185649367123936
#18563532227 13825578494871298734247872 #60614674261 13825578494371298734247872
#18563532228 13825678494871298734247872 #60614674262 13825678494371298734247872
#18563532229 27651356989742597468495744 #60614674263 27651356988742597468495744
#18563532230 27651356989742597468495745 = #60614674264 27651356989742597468495745
So we have a loop of length 42051142034. The smallest term in the loop appears to be 507434154592, so here is an abridged loop sequence (asterisk denotes the largest term; three twelve-digit local minima are also shown; indices of all these corrected February 29):
0 507434154592
1 1014868309184
2 2029736618368
3 2029736718368
4 4059473436736
5 8118946873472
6 8128946873472
7 16257893746944
8 16257893746945
9 32515787493890
10 65031574987780
11 65031574987880
12 65031574987890
13 130063149975780
14 131631410075780
15 13163141175780
16 13163141275780
17 26326282551560
18 26326282561560
19 52652565123120
20 105305130246240
... ...
17074586421 49512395802029907136051366345193519491458782692496790312698501120
17074586422 495123958020210007136051367345193519491458782692496790312698501220 *
17074586423 4951239580202117136051367345193519491458782692496790312698501230
... ...
25756695203 5007793970328
25756695204 517893970328
25756695205 1035787940656
... ...
25757984145 5097006463136
25757984146 509716463136
25757984147 1019432926272
... ...
27813217917 6806950060736
27813217918 680695160736
27813217919 1361390321472
... ...
42051142014 1128050902650182
42051142015 1228050902650182
42051142016 1238050902650182
42051142017 2476101805300364
42051142018 247610180531364
42051142019 495220361062728
42051142020 495230361062728
42051142021 990460722125456
42051142022 1000460723125456
42051142023 11460723125456
42051142024 12460723125456
42051142025 24921446250912
42051142026 24921456250912
42051142027 49842912501824
42051142028 99685825003648
42051142029 10068582513648
42051142030 1168582513648
42051142031 1268582513648
42051142032 2537165027296
42051142033 5074330054592
42051142034 507434154592
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ÉA
> Bravo and thanks, Hans!
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