A284900 - OEIS

%I #37 Nov 11 2022 04:35:11

%S 1,15,82,239,626,1230,2402,3823,6643,9390,14642,19598,28562,36030,

%T 51332,61167,83522,99645,130322,149614,196964,219630,279842,313486,

%U 391251,428430,538084,574078,707282,769980,923522,978671,1200644,1252830,1503652,1587677

%N a(n) = Sum_{d|n} (-1)^(n/d+1)*d^4.

%C Multiplicative because this sequence is the Dirichlet convolution of A000583 and A062157 which are both multiplicative. - _Andrew Howroyd_, Jul 20 2018

%H Seiichi Manyama, <a href="/A284900/b284900.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^4*x^k/(1 + x^k). - _Ilya Gutkovskiy_, Apr 07 2017

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

%F Multiplicative with a(2^e) = (7*2^(4*e+1)+1)/15, and a(p^e) = (p^(4*e+4) - 1)/(p^4 - 1) if p > 2.

%F Sum_{k=1..n} a(k) ~ c * n^5, where c = 3*zeta(5)/16 = 0.194423... . (End)

%t Table[Sum[(-1)^(n/d + 1)*d^4, {d, Divisors[n]}], {n, 50}] (* _Indranil Ghosh_, Apr 05 2017 *)

%t f[p_, e_] := (p^(4*e + 4) - 1)/(p^4 - 1); f[2, e_] := (7*2^(4*e + 1) + 1)/15; 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^4); \\ _Indranil Ghosh_, Apr 05 2017

%o (Python)

%o from sympy import divisors

%o print([sum([(-1)**(n//d + 1)*d**4 for d in divisors(n)]) for n in range(1, 51)]) # _Indranil Ghosh_, Apr 05 2017

%Y Sum_{d|n} (-1)^(n/d+1)*d^k: A000593 (k=1), A078306 (k=2), A078307 (k=3), this sequence (k=4), A284926 (k=5), A284927 (k=6), A321552 (k=7), A321553 (k=8), A321554 (k=9), A321555 (k=10), A321556 (k=11), A321557 (k=12).

%Y Cf. A000583, A013663, A062157.

%K nonn,mult

%O 1,2

%A _Seiichi Manyama_, Apr 05 2017