A338652 - OEIS

%I #30 Jan 08 2024 01:38:18

%S 0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,1,2,1,2,1,2,1,2,2,2,1,3,1,2,2,2,

%T 1,4,1,2,2,3,1,3,1,3,3,2,1,4,1,3,2,3,1,4,2,3,2,2,1,6,1,2,3,3,2,4,1,3,

%U 2,4,1,6,1,2,3,3,2,4,1,5,3,2,1,6,2,2,2,4,1,7,2,3,2,2,2,6,1,3,4,5,1,4,1,4,4,2,1,7,1,5

%N Number of divisors of n which are greater than 8.

%H Seiichi Manyama, <a href="/A338652/b338652.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Di#divisors">Index entries for sequences related to divisors of numbers</a>.

%F G.f.: Sum_{k>=1} x^(9*k) / (1 - x^k).

%F L.g.f.: -log( Product_{k>=9} (1 - x^k)^(1/k) ).

%F G.f.: Sum_{k>=9} x^k/(1 - x^k). - _Seiichi Manyama_, Jan 07 2023

%F Sum_{k=1..n} a(k) ~ n * (log(n) + 2*gamma - 1041/280), where gamma is Euler's constant (A001620). - _Amiram Eldar_, Jan 08 2024

%t Table[DivisorSum[n, 1 &, # > 8 &], {n, 1, 110}]

%t nmax = 110; CoefficientList[Series[Sum[x^(9 k)/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Drop[#, 1] &

%t nmax = 110; CoefficientList[Series[-Log[Product[(1 - x^k)^(1/k), {k, 9, nmax}]], {x, 0, nmax}], x] Range[0, nmax] // Drop[#, 1] &

%o (PARI) a(n) = sumdiv(n, d, d>8); \\ _Michel Marcus_, Apr 22 2021

%o (PARI) my(N=100, x='x+O('x^N)); concat([0, 0, 0, 0, 0, 0, 0, 0], Vec(sum(k=9, N, x^k/(1-x^k)))) \\ _Seiichi Manyama_, Jan 07 2023

%Y Column k=9 of A135539.

%Y Cf. A000005, A001620, A023645, A032741, A321014, A338648, A338649, A338650, A338651, A338653.

%K nonn,easy

%O 1,18

%A _Ilya Gutkovskiy_, Apr 22 2021

%E a(1)-a(8) prepended by _David A. Corneth_, Jun 13 2022