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%I #32 Nov 11 2022 04:34:39
%S 1,63,730,4031,15626,45990,117650,257983,532171,984438,1771562,
%T 2942630,4826810,7411950,11406980,16510911,24137570,33526773,47045882,
%U 62988406,85884500,111608406,148035890,188327590,244156251,304089030,387952660,474247150,594823322
%N a(n) = Sum_{d|n} (-1)^(n/d+1)*d^6.
%C Multiplicative because this sequence is the Dirichlet convolution of A001014 and A062157 which are both multiplicative. - _Andrew Howroyd_, Jul 20 2018
%H Seiichi Manyama, <a href="/A284927/b284927.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} k^6*x^k/(1 + x^k). - _Ilya Gutkovskiy_, Apr 07 2017
%F From _Amiram Eldar_, Nov 11 2022: (Start)
%F Multiplicative with a(2^e) = (31*2^(6*e+1)+1)/63, and a(p^e) = (p^(6*e+6) - 1)/(p^6 - 1) if p > 2.
%F Sum_{k=1..n} a(k) ~ c * n^7, where c = 9*zeta(7)/64 = 0.141799... . (End)
%t Table[Sum[(-1)^(n/d + 1)*d^6, {d, Divisors[n]}], {n, 50}] (* _Indranil Ghosh_, Apr 06 2017 *)
%t f[p_, e_] := (p^(6*e + 6) - 1)/(p^6 - 1); f[2, e_] := (31*2^(6*e + 1) + 1)/63; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* _Amiram Eldar_, Nov 11 2022 *)
%o (PARI) a(n) = sumdiv(n, d, (-1)^(n/d + 1)*d^6); \\ _Indranil Ghosh_, Apr 06 2017
%o (Python)
%o from sympy import divisors
%o print([sum([(-1)**(n//d + 1)*d**6 for d in divisors(n)]) for n in range(1, 51)]) # _Indranil Ghosh_, Apr 06 2017
%Y Sum_{d|n} (-1)^(n/d+1)*d^k: A000593 (k=1), A078306 (k=2), A078307 (k=3), A284900 (k=4), A284926 (k=5), this sequence (k=6), A321552 (k=7), A321553 (k=8), A321554 (k=9), A321555 (k=10), A321556 (k=11), A321557 (k=12).
%Y Cf. A001014, A013665, A062157.
%K nonn,mult
%O 1,2
%A _Seiichi Manyama_, Apr 06 2017
%E Keyword:mult added by _Andrew Howroyd_, Jul 23 2018
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