A080225 Number of perfect divisors of n.

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

Offset: 1

Author

Reinhard Zumkeller, Feb 07 2003

Keywords

• nonn

Comments

Number of divisors d of n with sigma(d) = 2*d (sigma = A000203).

Examples

			Divisors of n = 84: {1,2,3,4,6,7,12,14,21,24,28,42}, two of them are perfect: 6 = A000396(1) and 28 = A000396(2), therefore a(84) = 2.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Perfect Number.
• Wikipedia, Perfect number.

Crossrefs

Cf. A000203, A000396, A080224, A080226, A147645, A335118.

Programs

Haskell

a080225 n = length [d | d <- takeWhile (<= n) a000396_list, mod n d == 0]
-- Reinhard Zumkeller, Jan 20 2012

Mathematica

a[n_] := DivisorSum[n, 1 &, DivisorSigma[-1, #] == 2 &]; Array[a, 100] (* Amiram Eldar, Dec 31 2023 *)

PARI

a(n) = sumdiv(n, d, sigma(d, -1) == 2); \\ Amiram Eldar, Dec 31 2023

Formula

A080224(n) + a(n) + A080226(n) = A000005(n).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A335118 = 0.2045201... . - Amiram Eldar, Dec 31 2023