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%I #23 Mar 30 2019 08:37:37
%S 12801,348161,3225601,104988673,4294967297,7816642561,43796171521,
%T 49413980161,54745942917121,51125767490519041,18314818035992494081,
%U 18446744073709551617
%N Composite integers n such that n-1 divided by the binary period of 1/n (=A007733(n)) equals an integral power of 2.
%C All terms are odd. If even n belongs to this sequence, then n-1 is odd and thus (n-1)/A007733(n) is also odd and thus must be equal to 1. On the other hand, for even n, A007733(n) < n/2 <= n-1, i.e., (n-1)/A007733(n) > 1, a contradiction.
%C Subsequence of A001567.
%C Contains all composite Fermat numbers A000215(k) = 2^(2^k)+1 (which are composite for 5<=k<=32 and conjecturally for any k>=5). In particular, a(5) = A000215(5), a(12) = A000215(6), and a(13) <= A000215(7) = 2^128+1.
%C Pseudoprimes n such that (n-1)/ord_{n}(2) = 2^k for some k, where ord_{n}(2) = A002326((n-1)/2) is the multiplicative order of 2 mod n. Composite numbers n such that Od(ord_{n}(2)) = Od(n-1), where ord_{n}(2) as above and Od(m) = A000265(m) is the odd part of m. Note that if Od(ord_{n}(2)) = Od(n-1), then ord_{n}(2)|(n-1). - _Thomas Ordowski_, Mar 13 2019
%H Mathoverflow, <a href="http://mathoverflow.net/questions/168045/are-all-counterexamples-of-oeis-a226181-both-poulet-numbers-and-proth-numbers">Are all counterexamples of OEIS A226181 both Poulet numbers and Proth numbers?</a>
%e n = a(6) = 7816642561 = 2^15 * 238545 + 1 is the first term, which is not Proth number (A080075). The binary period of 1/n is 954180 = (n-1)/2^13.
%Y Cf. A000215, A001567, A007733, A226181.
%K nonn,more
%O 1,1
%A _Max Alekseyev_, May 29 2014
%E a(1)-a(3) from _Lear Young_; a(4)-a(5),a(9)-a(12) from _Max Alekseyev_; a(6),a(8) from Peter Kosinar; a(7) from _Chris Boyd_, May 29 2014.
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