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%I #18 Dec 31 2023 03:40:59
%S 1,2,1,2,2,2,2,2,1,3,1,3,1,3,3,2,1,2,1,3,2,3,1,3,2,3,1,3,1,4,1,2,1,2,
%T 4,3,1,2,1,4,1,3,1,3,3,2,1,3,2,3,2,3,1,2,2,3,2,2,1,5,1,2,2,2,2,3,1,2,
%U 1,6,1,3,1,2,3,2,3,3,1,4,1,2,1,4,2,2,1,3,1,4,2,3,1,2,2,3,1,3,1,4,1,3,1,3,5
%N Number of distinct generalized pentagonal numbers dividing n.
%C For more information about the generalized pentagonal numbers see A001318.
%H Amiram Eldar, <a href="/A244964/b244964.txt">Table of n, a(n) for n = 1..10000</a>
%F From _Amiram Eldar_, Dec 31 2023: (Start)
%F a(n) = Sum_{d|n} A080995(d).
%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 6 - 2*Pi/sqrt(3) = 2.372401... . (End)
%e For n = 10 the generalized pentagonal numbers <= 10 are [0, 1, 2, 5, 7]. There are three generalized pentagonal numbers that divide 10; they are [1, 2, 5], so a(10) = 3.
%t a[n_] := DivisorSum[n, 1 &, IntegerQ[Sqrt[24*# + 1]] &]; Array[a, 100] (* _Amiram Eldar_, Dec 31 2023 *)
%o (PARI) a(n) = sumdiv(n, d, issquare(24*d + 1)); \\ _Amiram Eldar_, Dec 31 2023
%Y Cf. A000005, A001221, A001318, A001511, A005086, A006519, A007862, A027750, A046951, A080995, A147645, A175003, A236103, A238442, A239930.
%K nonn
%O 1,2
%A _Omar E. Pol_, Jul 10 2014
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