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A321811 Sum of 7th powers of odd divisors of n.

%I A321811 #26 Feb 16 2025 08:33:57
%S A321811 1,1,2188,1,78126,2188,823544,1,4785157,78126,19487172,2188,62748518,
%T A321811 823544,170939688,1,410338674,4785157,893871740,78126,1801914272,
%U A321811 19487172,3404825448,2188,6103593751,62748518,10465138360,823544,17249876310
%N A321811 Sum of 7th powers of odd divisors of n.
%H A321811 Seiichi Manyama, <a href="/A321811/b321811.txt">Table of n, a(n) for n = 1..10000</a>
%H A321811 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 A321811 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/OddDivisorFunction.html">Odd Divisor Function</a>.
%H A321811 <a href="/index/Ge#Glaisher">Index entries for sequences mentioned by Glaisher</a>.
%F A321811 a(n) = A013955(A000265(n)) = sigma_7(odd part of n); in particular, a(2^k) = 1 for all k >= 0. - _M. F. Hasler_, Nov 26 2018
%F A321811 G.f.: Sum_{k>=1} (2*k - 1)^7*x^(2*k-1)/(1 - x^(2*k-1)). - _Ilya Gutkovskiy_, Dec 07 2018
%F A321811 From _Amiram Eldar_, Nov 02 2022: (Start)
%F A321811 Multiplicative with a(2^e) = 1 and a(p^e) = (p^(7*e+7)-1)/(p^7-1) for p > 2.
%F A321811 Sum_{k=1..n} a(k) ~ c * n^8, where c = zeta(8)/16 = Pi^8/151200 = 0.0627548... . (End)
%t A321811 a[n_] := DivisorSum[n, #^7 &, OddQ[#] &]; Array[a, 20] (* _Amiram Eldar_, Dec 07 2018 *)
%o A321811 (PARI) apply( A321811(n)=sigma(n>>valuation(n,2),7), [1..30]) \\ _M. F. Hasler_, Nov 26 2018
%o A321811 (Python)
%o A321811 from sympy import divisor_sigma
%o A321811 def A321811(n): return int(divisor_sigma(n>>(~n&n-1).bit_length(),7)) # _Chai Wah Wu_, Jul 16 2022
%Y A321811 Column k=7 of A285425.
%Y A321811 Cf. A050999, A051000, A051001, A051002, A321810 - A321816 (analog for 2nd .. 12th powers).
%Y A321811 Cf. A321543 - A321565, A321807 - A321836 for related sequences.
%Y A321811 Cf. A000265, A013666, A013955.
%K A321811 nonn,mult
%O A321811 1,3
%A A321811 _N. J. A. Sloane_, Nov 24 2018