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A321561 a(n) = Sum_{d divides n} (-1)^(d + n/d) * d^5. 3

%I

%S 1,-33,244,-993,3126,-8052,16808,-31713,59293,-103158,161052,-242292,

%T 371294,-554664,762744,-1014753,1419858,-1956669,2476100,-3104118,

%U 4101152,-5314716,6436344,-7737972,9768751,-12252702,14408200,-16690344,20511150

%N a(n) = Sum_{d divides n} (-1)^(d + n/d) * d^5.

%H G. C. Greubel, <a href="/A321561/b321561.txt">Table of n, a(n) for n = 1..1000</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^5*x^k/(1 + x^k). - _Ilya Gutkovskiy_, Nov 27 2018

%t a[n_] := DivisorSum[n, (-1)^(# + n/#)*#^5 &]; Array[a, 50] (* _Amiram Eldar_, Nov 27 2018 *)

%o (PARI) apply( A321561(n)=sumdiv(n, d, (-1)^(n\d-d)*d^5), [1..30]) \\ _M. F. Hasler_, Nov 26 2018

%o (MAGMA) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&+[(-1)^(k+1)*k^5*x^k/(1 + x^k) : k in [1..2*m]]) )); // _G. C. Greubel_, Nov 28 2018

%o (Sage) s=(sum((-1)^(k+1)*k^5*x^k/(1 + x^k) for k in (1..50))).series(x, 50); a = s.coefficients(x, sparse=False); a[1:] # _G. C. Greubel_, Nov 28 2018

%Y Column k=5 of A322083.

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

%K sign,mult

%O 1,2

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

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Last modified August 14 19:47 EDT 2020. Contains 336483 sequences. (Running on oeis4.)