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A376514 Number of divisors of n that are neither squarefree nor prime powers.

%I A376514 #6 Sep 27 2024 23:06:02
%S A376514 0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,1,0,0,0,2,0,0,0,1,0,0,0,0,0,0,
%T A376514 0,3,0,0,0,2,0,0,0,1,1,0,0,3,0,1,0,1,0,2,0,2,0,0,0,3,0,0,1,0,0,0,0,1,
%U A376514 0,0,0,5,0,0,1,1,0,0,0,3,0,0,0,3,0,0,0
%N A376514 Number of divisors of n that are neither squarefree nor prime powers.
%C A376514 This sequence is distinct from both A062977 and A335460.
%C A376514 A062977(36) = 0 while a(36) = 3 = card({12, 18, 36}).
%C A376514 A335460(60) = 6 while a(60) = 3 = card({12, 20, 60}).
%H A376514 Michael De Vlieger, <a href="/A376514/b376514.txt">Table of n, a(n) for n = 1..10000</a>
%F A376514 a(n) = tau(n) - 2^omega(n) - bigomega(n) + omega(n).
%F A376514 a(n) = A000005(n) - 2^A001221(n) - A001222(n) + A001221(n).
%F A376514 a(n) = A000005(n) - 2^A001221(n) - A046660(n).
%F A376514 Intersection of row n of A027750 and A126706.
%F A376514 a(n) > 0 for n in A126706.
%t A376514 {0}~Join~Table[DivisorSigma[0, n] - Total@ #1 - 2^#2 + #2 & @@ {#, Length[#]} &[FactorInteger[n][[All, -1]] ], {n, 2, 120}] (* or *)
%t A376514 Table[DivisorSum[n, 1 &, Nor[PrimePowerQ[#], SquareFreeQ[#]] &], {n, 120}]
%Y A376514 Cf. A000005, A001221, A001222, A027750, A046660, A126706, A376504.
%K A376514 nonn,easy
%O A376514 1,24
%A A376514 _Michael De Vlieger_, Sep 25 2024