A321824 - OEIS

%I #30 Aug 17 2024 14:25:42

%S 1,1,-6560,1,390626,-6560,-5764800,1,43040161,390626,-214358880,-6560,

%T 815730722,-5764800,-2562506560,1,6975757442,43040161,-16983563040,

%U 390626,37817088000,-214358880,-78310985280,-6560,152588281251,815730722

%N a(n) = Sum_{d|n, d==1 mod 4} d^8 - Sum_{d|n, d==3 mod 4} d^8.

%H Seiichi Manyama, <a href="/A321824/b321824.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 a(n) = a(A000265(n)). - _M. F. Hasler_, Nov 26 2018

%F G.f.: Sum_{k>=1} (-1)^(k-1)*(2*k - 1)^8*x^(2*k-1)/(1 - x^(2*k-1)). - _Ilya Gutkovskiy_, Dec 06 2018

%F Multiplicative with a(2^e) = 1, and for an odd prime p, ((p^8)^(e+1)-1)/(p^8-1) if p == 1 (mod 4) and ((-p^8)^(e+1)-1)/(-p^8-1) if p == 3 (mod 4). - _Amiram Eldar_, Sep 27 2023

%F a(n) = Sum_{d|n} d^8*sin(d*Pi/2). - _Ridouane Oudra_, Aug 17 2024

%t s[n_, r_] := DivisorSum[n, #^8 &, Mod[#, 4] == r &]; a[n_] := s[n, 1] - s[n, 3]; Array[a, 30] (* _Amiram Eldar_, Nov 26 2018 *)

%t f[p_, e_] := If[Mod[p, 4] == 1, ((p^8)^(e+1)-1)/(p^8-1), ((-p^8)^(e+1)-1)/(-p^8-1)]; f[2, e_] := 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 60] (* _Amiram Eldar_, Sep 27 2023 *)

%o (PARI) apply( A321824(n)=sumdiv(n>>valuation(n,2),d,(2-d%4)*d^8), [1..40]) \\ _M. F. Hasler_, Nov 26 2018

%Y Column k=8 of A322143.

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

%Y Cf. A000265.

%K sign,easy,mult

%O 1,3

%A _N. J. A. Sloane_, Nov 24 2018

%E Keyword mult from _Ilya Gutkovskiy_, Dec 06 2018