A294890 Number of divisors of n that are primitively abundant (A091191).

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 3, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 3

Offset: 1

Author

Antti Karttunen, Nov 14 2017

Keywords

• nonn

Comments

Records occur at 1, 12, 36, 60, 180, 420, 840, 2520, 7560, 9240, 24024, 60060, ... and they are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, ... Ten occurs for the first time as a(40040) = 10.

Examples

			Divisors of 24 are 1, 2, 3, 4, 6, 8, 12, 24. Only 12 is in A091191, thus a(24) = 1.
Divisors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36. Of these 12 and 18 are found in A091191, thus a(36) = 2.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..70560

Crossrefs

Cf. A080224, A091191, A294930.

Programs

Mathematica

q[n_] := Count[Divisors[n], ?(DivisorSigma[1, #] > 2*# &)] == 1; a[n] := DivisorSum[n, 1 &, q[#] &]; Array[a, 100] (* Amiram Eldar, Mar 14 2024 *)

PARI

A294937(n) = (sigma(n)>(2*n));
A294929(n) = sumdiv(n, d, (dA294937(d));
A294930(n) = (A294937(n)*(0==A294929(n)));
A294890(n) = sumdiv(n, d, A294930(d));

Formula

a(n) = Sum_{d|n} A294930(d).