A321547 - OEIS

%I #18 Nov 04 2022 10:47:21

%S 1,-255,6562,-65791,390626,-1673310,5764802,-16843007,43053283,

%T -99609630,214358882,-431720542,815730722,-1470024510,2563287812,

%U -4311810303,6975757442,-10978587165,16983563042,-25699675166,37828630724,-54661514910,78310985282,-110523811934,152588281251,-208011334110

%N a(n) = Sum_{d|n} (-1)^(d-1)*d^8.

%H Seiichi Manyama, <a href="/A321547/b321547.txt">Table of n, a(n) for n = 1..10000</a>

%H J. W. L. Glaisher, <a href="https://books.google.com/books?id=bLs9AQAAMAAJ&amp;pg=RA1-PA1">On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares</a>, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).

%H <a href="/index/Ge#Glaisher">Index entries for sequences mentioned by Glaisher</a>.

%F G.f.: Sum_{k>=1} (-1)^(k-1)*k^8*x^k/(1 - x^k). - _Ilya Gutkovskiy_, Dec 23 2018

%F Multiplicative with a(2^e) = 2 - (2^(8*e + 8) - 1)/255, and a(p^e) = (p^(8*e + 8) - 1)/(p^8 - 1) for p > 2. - _Amiram Eldar_, Nov 04 2022

%t f[p_, e_] := (p^(8*e + 8) - 1)/(p^8 - 1); f[2, e_] := 2 - (2^(8*e + 8) - 1)/255; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 24] (* _Amiram Eldar_, Nov 04 2022 *)

%o (PARI) apply( a(n)=sumdiv(n, d, (-1)^(d-1)*d^8), [1..30]) \\ _M. F. Hasler_, Nov 26 2018

%Y Cf. A321543 - A321565, A321807 - A321836 for similar sequences.

%K sign,mult

%O 1,2

%A _N. J. A. Sloane_, Nov 23 2018