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

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, 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, 0, 0, 0, 5, 0, 0, 1, 1, 0, 0, 0, 3, 0, 0, 0, 3, 0, 0, 0

Offset: 1

Author

Michael De Vlieger, Sep 25 2024

Keywords

• nonn
• easy

Comments

This sequence is distinct from both A062977 and A335460.

A062977(36) = 0 while a(36) = 3 = card({12, 18, 36}).

A335460(60) = 6 while a(60) = 3 = card({12, 20, 60}).

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A001221, A001222, A027750, A046660, A126706, A376504.

Programs

Mathematica

{0}~Join~Table[DivisorSigma[0, n] - Total@ #1 - 2^#2 + #2 & @@ {#, Length[#]} &[FactorInteger[n][[All, -1]] ], {n, 2, 120}] (* or *)
Table[DivisorSum[n, 1 &, Nor[PrimePowerQ[#], SquareFreeQ[#]] &], {n, 120}]

Formula

a(n) = tau(n) - 2^omega(n) - bigomega(n) + omega(n).

a(n) = A000005(n) - 2^A001221(n) - A001222(n) + A001221(n).

a(n) = A000005(n) - 2^A001221(n) - A046660(n).

Intersection of row n of A027750 and A126706.

a(n) > 0 for n in A126706.