A080225 - OEIS

%I #19 Dec 31 2023 06:23:18

%S 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,

%T 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,

%U 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

%N Number of perfect divisors of n.

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

%H Reinhard Zumkeller, <a href="/A080225/b080225.txt">Table of n, a(n) for n = 1..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PerfectNumber.html">Perfect Number</a>.

%H Wikipedia, <a href="http://www.wikipedia.org/wiki/Perfect_number">Perfect number</a>.

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

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

%e 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.

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

%o (Haskell)

%o a080225 n = length [d | d <- takeWhile (<= n) a000396_list, mod n d == 0]

%o -- _Reinhard Zumkeller_, Jan 20 2012

%o (PARI) a(n) = sumdiv(n, d, sigma(d, -1) == 2); \\ _Amiram Eldar_, Dec 31 2023

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

%K nonn

%O 1,84

%A _Reinhard Zumkeller_, Feb 07 2003