A381743 The number of divisors d of n such that d*n is abundant.

0, 0, 0, 0, 0, 3, 0, 0, 0, 2, 0, 6, 0, 1, 0, 0, 0, 6, 0, 6, 0, 0, 0, 8, 0, 0, 0, 5, 0, 8, 0, 0, 0, 0, 0, 9, 0, 0, 0, 8, 0, 8, 0, 4, 0, 0, 0, 10, 0, 3, 0, 4, 0, 8, 0, 8, 0, 0, 0, 12, 0, 0, 0, 0, 0, 8, 0, 2, 0, 8, 0, 12, 0, 0, 0, 2, 0, 8, 0, 10, 0, 0, 0, 12, 0, 0

Offset: 1

Author

Amiram Eldar, Mar 06 2025

Keywords

• nonn
• easy

Comments

The indices of records of this sequence are the highly composite numbers (A002182) without 2 and 4.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A000396, A002182, A005101, A341358, A381738, A381742.

Programs

Mathematica

a[n_] := DivisorSum[n, 1 &, DivisorSigma[-1, #*n] > 2 &]; Array[a, 100]

PARI

a(n) = sumdiv(n, d, sigma(d*n, -1) > 2);

Formula

a(n) = A000005(n) if and only if n is abundant (A005101).

a(n) = A000005(n) - 1 if and only if n is either 1, a perfect number (A000396), or in A341358.

a(n) >= 1 if and only if n is in A381738.

a(n) = 1 if and only if n is in A381742.