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%I #13 Jan 02 2023 12:30:54
%S 1,2,6,3,15,5,65,13,26,182,7,14,42,21,105,35,455,91,910,10,30,210,70,
%T 2730,39,78,546,273,1365,195,7995,41,82,246,123,615,205,2665,533,1066,
%U 11726,11,22,66,33,165,55,715,143,286,2002,77,154,462,231,1155,385,5005,1001,10010,110,330,2310,770,30030,429,858,6006,3003,15015,2145,87945,451,902
%N Suspected divisor-or-multiple permutation of squarefree numbers: a(n) = A019565(A304533(n)).
%C Each a(n) is always either a divisor or a multiple of a(n+1).
%C Consider A052330. Imagine that it is an automatic piano that "plays sequences" when an appropriate punched "tape" is fed to it (as its input), i.e., when it is composed from the right with an appropriate sequence p, as A019565(p(n)). The 1-bits in the binary expansion of each p(n) are the "holes" in the tape, and they determine which "tunes" are present on beat n. The "tunes" are actually "Fermi-Dirac primes" (A050376) that are multiplied together.
%C If the tape is constructed in such a way that between the successive beats (when moving from p(n) to p(n+1)), either a subset of 0-bits are toggled on (changed to 1's), or a subset of 1-bits are toggled off (changed to 0's), but no both kind of changes occur simultaneously, then when fed as an input to this piano, the resulting sequence "s" (the output) is guaranteed to satisfy the condition that s(n+1) is either a multiple or a divisor of s(n). Furthermore, if the given sequence p is itself a permutation of natural numbers, then also the produced sequence is. For example, Gray code A003188 and its inverse A006068 are such sequences, and when given as an "input tape" for A052330, they produce permutations A207901 and A302783.
%C There is a simpler instrument, called "squarefree piano" (A019565), with which it is possible to produce similar divisor-or-multiple sequences, but that contain only squarefree numbers. Given A003188 or A006068 as an input tape for it produces correspondingly sequences A302033 and A284003.
%C This sequence is obtained by playing "squarefree piano" with the same tape which yields A304531 when "Fermi-Dirac piano" is played with it. However, in this case the sequence A304531 is produced by a greedy algorithm, and thus its tape (A304533) is actually a back-formation, obtained from the "music" (A304531) by applying "tape-recorder" (A052331) to it. Note that this in not a subsequence of A304531, as the terms occur in different order than the squarefree terms of A304531.
%C See also Peter Munn's Apr 11 2018 message on SeqFan-mailing list.
%H Antti Karttunen, <a href="/A304537/b304537.txt">Table of n, a(n) for n = 0..8447</a>
%H Michel Marcus, Peter Munn, et al, <a href="http://list.seqfan.eu/oldermail/seqfan/2018-April/018581.html">Discussion on SeqFan-list</a>, April 2018
%F a(n) = A019565(A304533(n)) = A019565(A052331(A304531(1+n))).
%o (PARI)
%o A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
%o A304533(n) = A052331(A304531(1+n));
%o A304537(n) = A019565(A304533(n));
%Y Cf. A019565, A304531, A304533.
%Y Cf. also A303760, A303771.
%K nonn
%O 0,2
%A _Antti Karttunen_, May 15 2018
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