A294929 Number of proper divisors of n that are abundant (A005101).

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 3, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 6

Offset

1,36

Links

Antti Karttunen, Table of n, a(n) for n = 1..20000

Formula

a(n) = Sum_{d|n, d<n} A294937(d).

a(n) = A080224(n) - A294937(n).

a(n) + A294928(n) = A032741(n).

Example

The proper divisors of 24 are 1, 2, 3, 4, 6, 8, 12. Only one of these, 12, is abundant (in A005101), thus a(24) = 1.
The proper divisors of 120 are 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60. Six of these are abundant: 12, 20, 24, 30, 40, 60, thus a(120) = 6.

Mathematica

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

Prog

(PARI) A294929(n) = sumdiv(n, d, (d<n)*(sigma(d)>(2*d)));

Crossrefs

Cf. A005101, A032741, A080224, A294889, A294926, A294927, A294928, A294930, A294937.

Cf. also A294902.

Sequence in context: A062977 A357879 A072325 * A076948 A255309 A335446

Adjacent sequences: A294926 A294927 A294928 * A294930 A294931 A294932

Keyword

nonn

Author

Antti Karttunen, Nov 14 2017

Status

approved