A321553 a(n) = Sum_{d|n} (-1)^(n/d+1)*d^8.
1, 255, 6562, 65279, 390626, 1673310, 5764802, 16711423, 43053283, 99609630, 214358882, 428360798, 815730722, 1470024510, 2563287812, 4278124287, 6975757442, 10978587165, 16983563042, 25499674654, 37828630724, 54661514910, 78310985282, 109660357726, 152588281251, 208011334110, 282472589764
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
-
Mathematica
Table[Total[(-1)^(n/#+1) #^8&/@Divisors[n]],{n,30}] (* Harvey P. Dale, May 05 2021 *) f[p_, e_] := (p^(8*e + 8) - 1)/(p^8 - 1); f[2, e_] := (127*2^(8*e + 1) + 1)/255; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 11 2022 *) -
PARI
apply( A321553(n)=sumdiv(n, d, (-1)^(n\d-1)*d^8), [1..30]) \\ M. F. Hasler, Nov 26 2018
Formula
G.f.: Sum_{k>=1} k^8*x^k/(1 + x^k). - Seiichi Manyama, Nov 23 2018
From Amiram Eldar, Nov 11 2022: (Start)
Multiplicative with a(2^e) = (127*2^(8*e+1)+1)/255, and a(p^e) = (p^(8*e+8) - 1)/(p^8 - 1) if p > 2.
Sum_{k=1..n} a(k) ~ c * n^9, where c = 85*zeta(9)/768 = 0.110899... . (End)