A321829 - OEIS

%I #18 Nov 05 2023 09:03:23

%S 1,32,242,1024,3126,7744,16806,32768,58807,100032,161050,247808,

%T 371294,537792,756492,1048576,1419858,1881824,2476098,3201024,4067052,

%U 5153600,6436342,7929856,9768751,11881408,14290100,17209344,20511150,24207744,28629150,33554432,38974100,45435456,52535556

%N a(n) = Sum_{d|n, n/d==1 mod 4} d^5 - Sum_{d|n, n/d==3 mod 4} d^5.

%H Seiichi Manyama, <a href="/A321829/b321829.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} k^5*x^k/(1 + x^(2*k)). - _Ilya Gutkovskiy_, Nov 26 2018

%F Multiplicative with a(p^e) = round(p^(5e+5)/(p^5 + p%4 - 2)), where p%4 is the remainder of p modulo 4. (Following R. Israel in A321833.) - _M. F. Hasler_, Nov 26 2018

%F Sum_{k=1..n} a(k) ~ c * n^6 / 6, where c = A175570. - _Amiram Eldar_, Nov 04 2023

%t s[n_,r_] := DivisorSum[n, # ^5 &, Mod[n/#,4]==r &]; a[n_] := s[n,1] - s[n,3]; Array[a, 30] (* _Amiram Eldar_, Nov 26 2018 *)

%t s[n_] := If[OddQ[n], (-1)^((n-1)/2), 0]; (* A101455 *)

%t f[p_, e_] := (p^(5*e+5) - s[p]^(e+1))/(p^5 - s[p]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Nov 04 2023 *)

%o (PARI) apply( A321829(n)=factorback(apply(f->f[1]^(5*f[2]+5)\/(f[1]^5+f[1]%4-2),Col(factor(n)))), [1..40]) \\ _M. F. Hasler_, Nov 26 2018

%Y Cf. A101455, A175570.

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

%Y Glaisher's E'_i (i=0..12): A002654, A050469, A050470, A050471, A050468, this sequence, A321830, A321831, A321832, A321833, A321834, A321835, A321836.

%K nonn,easy,mult

%O 1,2

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