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%I #19 Jul 17 2021 07:00:56
%S 1,16,32,64,112,128,256,395,448,512,1024,1093,1168,1368,1472,1792,
%T 2013,2048,3279,3344,3511,3968,4096,5472,5696,7168,7651,8192,10533,
%U 14209,16384,17488,19674,21672,21888,22953,27552,28672,31599,32768,33883,34905,34976
%N Numbers k such that k^2 | A038199(k).
%C Serret proved in 1855 a generalization of Fermat's little theorem: for b >= 1, Sum_{d|k} mu(d)*b^(k/d) == 0 (mod k). This sequence includes numbers k such that k^2 divides the sum with base b=2.
%C Includes all the powers of 2 above 8.
%C An alternative generalization of Wieferich primes (A001220) which are the prime terms of this sequence.
%C Also numbers k such that k | A059966(k).
%D Wacław Sierpiński, Elementary Theory of Numbers, Elsevier, North Holland, 1988, page 217.
%H Amiram Eldar, <a href="/A317475/b317475.txt">Table of n, a(n) for n = 1..250</a>
%H Joseph-Alfred Serret, <a href="http://www.numdam.org/item/NAM_1855_1_14__261_1">Théorème de Fermat généralisé</a>, Nouvelles Annales de Mathématiques, Vol. 14 (1855), pp. 261-262.
%e 16 is in the sequence since Sum_{d|16} mu(d)*2^(16/d) = 65280 = 255 * 16^2.
%t f[n_] := DivisorSum[n, MoebiusMu[#] * 2^(n/#) &]; Select[Range[1000], Divisible[f[#], #^2] &]
%o (PARI) isok(n) = frac(sumdiv(n, d, moebius(n/d)*(2^d-1))/n^2) == 0; \\ _Michel Marcus_, Jul 30 2018
%Y Cf. A000079, A001220, A038199, A059966, A077816, A182297.
%K nonn
%O 1,2
%A _Amiram Eldar_, Jul 29 2018
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