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Number of square pyramidal numbers dividing n.
4

%I #13 Jan 02 2024 02:47:30

%S 1,1,1,1,2,1,1,1,1,2,1,1,1,2,2,1,1,1,1,2,1,1,1,1,2,1,1,2,1,3,1,1,1,1,

%T 2,1,1,1,1,2,1,2,1,1,2,1,1,1,1,2,1,1,1,1,3,2,1,1,1,3,1,1,1,1,2,1,1,1,

%U 1,3,1,1,1,1,2,1,1,1,1,2,1,1,1,2,2,1,1,1,1,3,2,1,1,1,2,1,1,2,1,2,1,1,1,1,2,1,1,1,1,3,1,2,1,1,2,1,1,1,1,3

%N Number of square pyramidal numbers dividing n.

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

%H <a href="/index/Ps#pyramidal_numbers">Index to sequences related to pyramidal numbers</a>.

%F G.f.: Sum_{k>=1} x^(k*(k+1)*(2*k+1)/6)/(1 - x^(k*(k+1)*(2*k+1)/6)).

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 18 - 24*log(2) = 1.364467... . - _Amiram Eldar_, Jan 02 2024

%e a(10) = 2 because 10 has 4 divisors {1,2,5,10} among which 2 divisors {1,5} are square pyramidal numbers.

%t Rest[CoefficientList[Series[Sum[x^(k (k + 1) (2 k + 1)/6)/(1 - x^(k (k + 1) (2 k + 1)/6)), {k, 120}], {x, 0, 120}], x]]

%Y Cf. A000330, A046951.

%K nonn,easy

%O 1,5

%A _Ilya Gutkovskiy_, Dec 13 2016