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A316860 Integers k that do not divide A053818(k).

%I A316860 #23 Nov 15 2020 22:33:48
%S A316860 2,3,4,6,8,9,12,15,16,18,24,27,30,32,33,36,45,48,51,54,60,64,66,69,72,
%T A316860 75,81,87,90,96,99,102,108,120,123,128,132,135,138,141,144,150,153,
%U A316860 159,162,165,174,177,180,192,198,204,207,213,216,225,240,243,246,249
%N A316860 Integers k that do not divide A053818(k).
%C A316860 k is a term iff k = 2^e or k = 3^e*Product_{i=1..s} p_i^e_i, p_i == 2 (mod 3) and e >= 1. If k = 2^e, A053818(k) == (1/2)*k (mod k); if k = 3^e*Product_{i=1..s} p_i^e_i, A053818(k) == (2/3)*k for even s and (1/3)*k for odd s. - Corrected by _Robert Israel_, Nov 15 2020
%C A316860 Terms < N are getting more and more sparse as N increases. The number of terms below 100, 1000, 10000 and 100000 are 31, 187, 1431 and 12059, respectively.
%H A316860 Jianing Song, <a href="/A316860/b316860.txt">Table of n, a(n) for n = 1..12059</a> (all terms below 100000)
%e A316860 A053818(16) mod 16 = 680 mod 16 = 8 != 0, so 16 is a term.
%e A316860 A053818(33) mod 33 = 7370 mod 33 = 11 != 0, so 33 is a term.
%e A316860 A053818(21) mod 21 = 1806 mod 21 = 0, so 21 is not a term.
%t A316860 Select[Range@ 250, Function[n, Mod[Total[Select[Range@ n, GCD[#, n] == 1 &]^2], n] != 0]] (* _Michael De Vlieger_, Jul 19 2018 *)
%o A316860 (PARI) pr(n)=my(f=factor(n)[, 1]); prod(i=1, #f, abs(f[i]%3-1));
%o A316860 for(n=2, 1000, if(omega(2*n)==1, print1(n, ", "), if(n%3==0&&pr(n), print1(n, ", "))))
%Y A316860 Cf. A053818.
%K A316860 nonn,easy
%O A316860 1,1
%A A316860 _Jianing Song_, Jul 15 2018