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a(n) = Sum_{d|n, n/d odd} d^6 for n > 0.
3

%I #23 Jan 09 2023 01:50:38

%S 1,64,730,4096,15626,46720,117650,262144,532171,1000064,1771562,

%T 2990080,4826810,7529600,11406980,16777216,24137570,34058944,47045882,

%U 64004096,85884500,113379968,148035890,191365120,244156251,308915840,387952660

%N a(n) = Sum_{d|n, n/d odd} d^6 for n > 0.

%H Seiichi Manyama, <a href="/A321817/b321817.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^6*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^(6*e) and a(p^e) = (p^(6*e+6)-1)/(p^6-1) for p > 2.

%F Sum_{k=1..n} a(k) ~ c * n^7, where c = 127*zeta(7)/896 = 0.142924... . (End)

%F Dirichlet g.f.: zeta(s)*zeta(s-6)*(1-1/2^s). - _Amiram Eldar_, Jan 08 2023

%t a[n_] := DivisorSum[n, #^6 &, OddQ[n/#] &]; Array[a, 30] (* _Amiram Eldar_, Nov 02 2022 *)

%o (PARI) apply( A321817(n)=sumdiv(n,d,if(bittest(n\d,0),d^6)), [1..30]) \\ _M. F. Hasler_, Nov 26 2018

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

%Y Cf. A013665.

%K nonn,mult

%O 1,2

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