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

1,84

Comments

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

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Perfect Number.

Wikipedia, Perfect number.

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

Example

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.

Mathematica

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

Prog

(Haskell)
a080225 n = length [d | d <- takeWhile (<= n) a000396_list, mod n d == 0]
-- Reinhard Zumkeller, Jan 20 2012
(PARI) a(n) = sumdiv(n, d, sigma(d, -1) == 2); \\ Amiram Eldar, Dec 31 2023

Crossrefs

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

Sequence in context: A317576 A070205 A138363 * A122841 A326075 A060862

Adjacent sequences: A080222 A080223 A080224 * A080226 A080227 A080228

Keyword

nonn

Author

Reinhard Zumkeller, Feb 07 2003

Status

approved