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A321810 Sum of 6th powers of odd divisors of n.

%I A321810 #27 Feb 16 2025 08:33:57
%S A321810 1,1,730,1,15626,730,117650,1,532171,15626,1771562,730,4826810,117650,
%T A321810 11406980,1,24137570,532171,47045882,15626,85884500,1771562,148035890,
%U A321810 730,244156251,4826810,387952660,117650,594823322,11406980,887503682
%N A321810 Sum of 6th powers of odd divisors of n.
%H A321810 Seiichi Manyama, <a href="/A321810/b321810.txt">Table of n, a(n) for n = 1..10000</a>
%H A321810 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 A321810 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/OddDivisorFunction.html">Odd Divisor Function</a>.
%H A321810 <a href="/index/Ge#Glaisher">Index entries for sequences mentioned by Glaisher</a>.
%F A321810 a(n) = A013954(A000265(n)) = sigma_6(odd part of n); in particular, a(2^k) = 1 for all k >= 0. - _M. F. Hasler_, Nov 26 2018
%F A321810 G.f.: Sum_{k>=1} (2*k - 1)^6*x^(2*k-1)/(1 - x^(2*k-1)). - _Ilya Gutkovskiy_, Dec 22 2018
%F A321810 From _Amiram Eldar_, Nov 02 2022: (Start)
%F A321810 Multiplicative with a(2^e) = 1 and a(p^e) = (p^(6*e+6)-1)/(p^6-1) for p > 2.
%F A321810 Sum_{k=1..n} a(k) ~ c * n^7, where c = zeta(7)/14 = 0.0720249... . (End)
%F A321810 a(n) + a(n/2)*2^6 = A013954(n) where a(.)=0 for non-integer arguments. - _R. J. Mathar_, Aug 15 2023
%t A321810 f[2, e_] := 1; f[p_, e_] := (p^(6*e + 6) - 1)/(p^6 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Nov 02 2022 *)
%o A321810 (PARI) apply( A321810(n)=sigma(n>>valuation(n,2),6), [1..30]) \\ _M. F. Hasler_, Nov 26 2018
%o A321810 (Python)
%o A321810 from sympy import divisor_sigma
%o A321810 def A321810(n): return int(divisor_sigma(n>>(~n&n-1).bit_length(),6)) # _Chai Wah Wu_, Jul 16 2022
%Y A321810 Column k=6 of A285425.
%Y A321810 Cf. A050999, A051000, A051001, A051002, A321811 - A321816 (analog for 2nd .. 12th powers).
%Y A321810 Cf. A321543 - A321565, A321807 - A321836 for related sequences.
%Y A321810 Cf. A000265, A013665, A013954.
%K A321810 nonn,mult
%O A321810 1,3
%A A321810 _N. J. A. Sloane_, Nov 24 2018