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%I #43 Jun 24 2022 05:29:44
%S 2,3,14,32,60,96,120,128,132,244,264,388,480,484,488,2064,1056,571,
%T 776,960,968,976,980,2112,2128,1143,1536,1552,1556,1920,3872,1937,
%U 3904,3920,1961,4128,4256,3104,6224,3113,3844,3848,7808,7824,7840,8256,8448,8452
%N a(n) is the smallest index k such that prime(n) divides both A090252(k) and A090252(2*k+1).
%C For n > 2, a(n) is not the smallest k such that prime(n) divides A090252(k), but it is the smallest k such that prime(n) divides both A090252(k) and A090252(2*k+1). If k_(0) = a(n) we may find either an infinite or finite range of indices where prime(n) divides A090252 using the recurrence k_(n) = 2*k_(n-1)+1, but there is a caveat: in very rare cases, some k values of this recurrence may be wrong by +-1, and the next iteration will then fit again. This uncertainty is caused by the fact that two terms of A090252 will be governed by the same floor(n/2) history. For yet unknown reasons, there may be an upper limit where such a recurrence may break.
%C This works because in A090252 the number of primes which do not divide the last floor(n/2) terms is growing faster than they are used up by this sequence. For each prime p then there exists an index k into A090252 where the supply of unused factors is so large that, when p becomes coprime to the last floor(n/2) terms, we can always immediately find a matching second prime to build a yet-unused semiprime or use p as a yet-unused power of itself.
%H Michael S. Branicky, <a href="/A355176/b355176.txt">Table of n, a(n) for n = 1..476</a>
%F A090252(a(n)) mod A000040(n) = 0 and a(n) is either even or A090252((a(n)-1)/2) mod A000040(n) > 0 is valid too.
%F A090252(2*a(n)+1) mod A000040(n) = 0.
%F A090252(f^m(a(n))) mod A000040(n) = 0, with f(x) = 2*x+1. The range of m is yet unknown.
%e prime(1) = 2 divides A090252(2) = 2, A090252(5) = 4, A090252(11) = 8, A090252(23) = 16, A090252(47) = 26, ... .
%e 2*2+1 = 5; 2*5+1 = 11; 2*11+1 = 23; 2*23+1 = 47.
%Y Cf. A000040, A090252.
%K nonn
%O 1,1
%A _Thomas Scheuerle_, Jun 22 2022
%E a(41) and beyond (using _Russ Cox_'s gzipped b-file at A090252) from _Michael S. Branicky_, Jun 23 2022
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