%I #22 Nov 11 2022 04:34:32

%S 1,127,2188,16255,78126,277876,823544,2080639,4785157,9922002,

%T 19487172,35565940,62748518,104590088,170939688,266321791,410338674,

%U 607714939,893871740,1269938130,1801914272,2474870844,3404825448,4552438132,6103593751,7969061786,10465138360,13386707720,17249876310

%N a(n) = Sum_{d|n} (-1)^(n/d+1)*d^7.

%H Seiichi Manyama, <a href="/A321552/b321552.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^7*x^k/(1 + x^k). - _Seiichi Manyama_, Nov 23 2018

%F From _Amiram Eldar_, Nov 11 2022: (Start)

%F Multiplicative with a(2^e) = (63*2^(7*e+1)+1)/127, and a(p^e) = (p^(7*e+7) - 1)/(p^7 - 1) if p > 2.

%F Sum_{k=1..n} a(k) ~ c * n^8, where c = 127*zeta(8)/1024 = 0.124529... . (End)

%t f[p_, e_] := (p^(7*e + 7) - 1)/(p^7 - 1); f[2, e_] := (63*2^(7*e + 1) + 1)/127; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* _Amiram Eldar_, Nov 11 2022 *)

%o (PARI) apply( A321552(n)=sumdiv(n, d, (-1)^(n\d-1)*d^7), [1..30]) \\ _M. F. Hasler_, Nov 26 2018

%Y Sum_{k>=1} k^b*x^k/(1 + x^k): A000593 (b=1), A078306 (b=2), A078307 (b=3), A284900 (b=4), A284926 (b=5), A284927 (b=6), this sequence (b=7), A321553 (b=8), A321554 (b=9), A321555 (b=10), A321556 (b=11), A321557 (b=12).

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

%Y Cf. A013666.

%K nonn,mult

%O 1,2

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