|
%I #23 Nov 04 2022 10:47:33
%S 1,-31,244,-1055,3126,-7564,16808,-33823,59293,-96906,161052,-257420,
%T 371294,-521048,762744,-1082399,1419858,-1838083,2476100,-3297930,
%U 4101152,-4992612,6436344,-8252812,9768751,-11510114,14408200,-17732440,20511150,-23645064,28629152,-34636831,39296688,-44015598
%N a(n) = Sum_{d|n} (-1)^(d-1)*d^5.
%H Seiichi Manyama, <a href="/A321544/b321544.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&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_, Dec 23 2018
%F G.f.: Sum_{n >= 1} x^n*(x^(4*n) - 26*x^(3*n) + 66*x^(2*n) - 26*x^n + 1)/(1 + x^n)^6 (note [1,26,66,26,1] is row 5 of A008292). - _Peter Bala_, Jan 11 2021
%F Multiplicative with a(2^e) = 2 - (2^(5*e + 5) - 1)/31, and a(p^e) = (p^(5*e + 5) - 1)/(p^5 - 1) for p > 2. - _Amiram Eldar_, Nov 04 2022
%p with(numtheory):
%p a := n -> add( (-1)^(d-1)*d^5, d in divisors(n) ): seq(a(n), n = 1..40);
%p # _Peter Bala_, Jan 11 2021
%t f[p_, e_] := (p^(5*e + 5) - 1)/(p^5 - 1); f[2, e_] := 2 - (2^(5*e + 5) - 1)/31; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 35] (* _Amiram Eldar_, Nov 04 2022 *)
%o (PARI) apply( a(n)=sumdiv(n, d, (-1)^(d-1)*d^5), [1..30]) \\ _M. F. Hasler_, Nov 26 2018
%Y Divisor sums Sum_{d|n} (-1)^(d-1)*d^k: A048272 (k = 0), A002129 (k = 1), A321543 (k = 2), A138503 (k = 3), A279395 (k = 4, unsigned), A321545 - A321551 (k = 6 to k = 12).
%Y Cf. A321552 - A321565, A321807 - A321836 for similar sequences.
%K sign,mult
%O 1,2
%A _N. J. A. Sloane_, Nov 23 2018
|
