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A080225
Number of perfect divisors of n.
8
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
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
KEYWORD
nonn
AUTHOR
Reinhard Zumkeller, Feb 07 2003
STATUS
approved