A321820 - OEIS

%I #26 Jan 09 2023 01:50:14

%S 1,4096,531442,16777216,244140626,2176786432,13841287202,68719476736,

%T 282430067923,1000000004096,3138428376722,8916117225472,

%U 23298085122482,56693912379392,129746582562692,281474976710656,582622237229762,1156833558212608

%N a(n) = Sum_{d|n, n/d odd} d^12 for n > 0.

%H Seiichi Manyama, <a href="/A321820/b321820.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^12*x^k/(1 - x^(2*k)). - _Ilya Gutkovskiy_, Dec 22 2018

%F From _Amiram Eldar_, Nov 02 2022: (Start)

%F Multiplicative with a(2^e) = 2^(12*e) and a(p^e) = (p^(12*e+12)-1)/(p^12-1) for p > 2.

%F Sum_{k=1..n} a(k) ~ c * n^13, where c = 8191*zeta(13)/106496 = 0.0769231... . (End)

%F Dirichlet g.f.: zeta(s)*zeta(s-12)*(1-1/2^s). - _Amiram Eldar_, Jan 09 2023

%t a[n_] := DivisorSum[n, #^12 &, OddQ[n/#] &]; Array[a, 20] (* _Amiram Eldar_, Nov 02 2022 *)

%o (PARI) apply( A321820(n)=sumdiv(n,d,if(bittest(n\d,0),d^12)), [1..30]) \\ _M. F. Hasler_, Nov 26 2018

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

%Y Cf. A013671.

%K nonn,mult

%O 1,2

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