A321831 - OEIS

%I #21 Sep 27 2024 16:22:35

%S 1,128,2186,16384,78126,279808,823542,2097152,4780783,10000128,

%T 19487170,35815424,62748518,105413376,170783436,268435456,410338674,

%U 611940224,893871738,1280016384,1800262812,2494357760,3404825446,4584374272,6103593751

%N a(n) = Sum_{d|n, n/d==1 mod 4} d^7 - Sum_{d|n, n/d==3 mod 4} d^7.

%H Seiichi Manyama, <a href="/A321831/b321831.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^(2*k)). - _Ilya Gutkovskiy_, Nov 26 2018

%F Multiplicative with a(p^e) = round(p^(7e+7)/(p^7 + p%4 - 2)), where p%4 is the remainder of p modulo 4. (Following R. Israel in A321833.) - _M. F. Hasler_, Nov 26 2018

%F Sum_{k=1..n} a(k) ~ c * n^8 / 8, where c = A258815. - _Amiram Eldar_, Nov 04 2023

%F a(n) = Sum_{d|n} (n/d)^7*sin(d*Pi/2). - _Ridouane Oudra_, Sep 27 2024

%t s[n_,r_] := DivisorSum[n, #^7 &, Mod[n/#,4]==r &]; a[n_] := s[n,1] - s[n,3]; Array[a, 30] (* _Amiram Eldar_, Nov 26 2018 *)

%t s[n_] := If[OddQ[n], (-1)^((n-1)/2), 0]; (* A101455 *)

%t f[p_, e_] := (p^(7*e+7) - s[p]^(e+1))/(p^7 - s[p]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Nov 04 2023 *)

%o (PARI) apply( A321831(n)=factorback(apply(f->f[1]^(7*f[2]+7)\/(f[1]^7+f[1]%4-2),Col(factor(n)))), [1..30]) \\ _M. F. Hasler_, Nov 26 2018

%Y Cf. A101455, A258815.

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

%Y Glaisher's E'_i (i=0..12): A002654, A050469, A050470, A050471, A050468, A321829, A321830, this sequence, A321832, A321833, A321834, A321835, A321836.

%K nonn,easy,mult

%O 1,2

%A _N. J. A. Sloane_, Nov 24 2018