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

1,36

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.

Links

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

Formula

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

Example

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.

Mathematica

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

Prog

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

Crossrefs

Cf. A080224, A091191, A294930.

Sequence in context: A156062 A156064 A158757 * A325562 A176917 A302110

Adjacent sequences: A294887 A294888 A294889 * A294891 A294892 A294893

Keyword

nonn

Author

Antti Karttunen, Nov 14 2017

Status

approved