A294890 - OEIS

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

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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));