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%I #8 Jan 11 2019 09:57:59
%S 9,15,25,27,33,39,49,55,57,63,81,87,95,111,119,121,125,135,143,153,
%T 159,161,169,175,177,183,201,207,209,225,243,249,287,289,295,297,303,
%U 319,321,329,335,343,351,361,369,375,391,393,407,415,417,423,447,489,497
%N Odd composite numbers n for which A137576((n-1)/2)-1 is divisible by phi(n).
%C If p is an odd prime then A137576((p-1)/2)=p. Therefore the composite numbers n may be considered as quasiprimes. In particular, if (m,n)=1 we have a natural generalization of the little Fermat theorem: m^(A137576((n-1)/ 2)-1)=1 mod n.
%H Ray Chandler, <a href="/A138193/b138193.txt">Table of n, a(n) for n=1..1239</a>
%e a(1)=9: A137576(4)=13 and 13-1 is divisible by phi(9)=6.
%t A137576[n_] := Module[{t}, (t = MultiplicativeOrder[2, 2 n + 1])* DivisorSum[2 n + 1, EulerPhi[#]/MultiplicativeOrder[2, #] &] - t + 1];
%t okQ[n_] := OddQ[n] && CompositeQ[n] && Divisible[A137576[(n - 1)/2] - 1, EulerPhi[n]];
%t Reap[For[k = 1, k < 500, k += 2, If[okQ[k], Print[k]; Sow[k]]]][[2, 1]] (* _Jean-François Alcover_, Jan 11 2019 *)
%Y Cf. A137576, A002326, A006694.
%K nonn
%O 1,1
%A _Vladimir Shevelev_, May 04 2008
%E Extended by _Ray Chandler_, May 08 2008
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