A321818 a(n) = Sum_{d|n, n/d odd} d^8 for n > 0.
1, 256, 6562, 65536, 390626, 1679872, 5764802, 16777216, 43053283, 100000256, 214358882, 430047232, 815730722, 1475789312, 2563287812, 4294967296, 6975757442, 11021640448, 16983563042, 25600065536, 37828630724, 54875873792, 78310985282
Offset: 1
Links
- Seiichi Manyama, Table of n, a(n) for n = 1..10000
- J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
- Index entries for sequences mentioned by Glaisher.
Programs
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Mathematica
a[n_] := DivisorSum[n, #^8 &, OddQ[n/#] &]; Array[a, 24] (* Amiram Eldar, Nov 02 2022 *)
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PARI
apply( A321818(n)=sumdiv(n,d,if(bittest(n\d,0),d^8)), [1..30]) \\ M. F. Hasler, Nov 26 2018
Formula
G.f.: Sum_{k>=1} k^8*x^k/(1 - x^(2*k)). - Ilya Gutkovskiy, Dec 22 2018
From Amiram Eldar, Nov 02 2022: (Start)
Multiplicative with a(2^e) = 2^(8*e) and a(p^e) = (p^(8*e+8)-1)/(p^8-1) for p > 2.
Sum_{k=1..n} a(k) ~ c * n^9, where c = 511*zeta(9)/4608 = 0.1111168... . (End)
Dirichlet g.f.: zeta(s)*zeta(s-8)*(1-1/2^s). - Amiram Eldar, Jan 09 2023