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A138217 Odd numbers n for which A137576((n-1)/2)-1 is a multiple of A000010(n).

%I A138217 #10 Jan 11 2019 09:58:05
%S A138217 1,3,5,7,9,11,13,15,17,19,23,25,27,29,31,33,37,39,41,43,47,49,53,55,
%T A138217 57,59,61,63,67,71,73,79,81,83,87,89,95,97,101,103,107,109,111,113,
%U A138217 119,121,125,127,131,135,137,139,143,149,151,153,157,159,161,163,167,169,173
%N A138217 Odd numbers n for which A137576((n-1)/2)-1 is a multiple of A000010(n).
%C A138217 The sequence contains all odd primes. Indeed, if p is a prime then A137576((p-1)/2)-1=p-1=A000010(p).
%C A138217 Conjecture: the sequence contains infinitely many composite numbers.
%C A138217 The conjecture is true because of the sequence contains all powers of odd primes. Indeed, A137576((P^k-1)/2)-1=k*A000010(p^k). - _Vladimir Shevelev_, May 29 2008
%H A138217 Ray Chandler, <a href="/A138217/b138217.txt">Table of n, a(n) for n=1..3501</a>
%t A138217 A137576[n_] := Module[{t}, (t = MultiplicativeOrder[2, 2 n + 1])* DivisorSum[2 n + 1, EulerPhi[#]/MultiplicativeOrder[2, #] &] - t + 1];
%t A138217 okQ[n_] := OddQ[n] && Divisible[A137576[(n - 1)/2] - 1, EulerPhi[n]];
%t A138217 Reap[For[k = 1, k < 200, k += 2, If[okQ[k], Sow[k]]]][[2, 1]] (* _Jean-François Alcover_, Jan 11 2019 *)
%Y A138217 Cf. A137576, A000010, A002326, A006694.
%K A138217 nonn
%O A138217 1,2
%A A138217 _Vladimir Shevelev_, May 05 2008
%E A138217 Extended by _Ray Chandler_, May 08 2008