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Sum of 6th powers of odd divisors of n.
11

%I #26 Aug 15 2023 13:29:17

%S 1,1,730,1,15626,730,117650,1,532171,15626,1771562,730,4826810,117650,

%T 11406980,1,24137570,532171,47045882,15626,85884500,1771562,148035890,

%U 730,244156251,4826810,387952660,117650,594823322,11406980,887503682

%N Sum of 6th powers of odd divisors of n.

%H Seiichi Manyama, <a href="/A321810/b321810.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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/OddDivisorFunction.html">Odd Divisor Function</a>.

%H <a href="/index/Ge#Glaisher">Index entries for sequences mentioned by Glaisher</a>.

%F 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 G.f.: Sum_{k>=1} (2*k - 1)^6*x^(2*k-1)/(1 - x^(2*k-1)). - _Ilya Gutkovskiy_, Dec 22 2018

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

%F Multiplicative with a(2^e) = 1 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 = zeta(7)/14 = 0.0720249... . (End)

%F a(n) + a(n/2)*2^6 = A013954(n) where a(.)=0 for non-integer arguments. - _R. J. Mathar_, Aug 15 2023

%t 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 (PARI) apply( A321810(n)=sigma(n>>valuation(n,2),6), [1..30]) \\ _M. F. Hasler_, Nov 26 2018

%o (Python)

%o from sympy import divisor_sigma

%o def A321810(n): return int(divisor_sigma(n>>(~n&n-1).bit_length(),6)) # _Chai Wah Wu_, Jul 16 2022

%Y Column k=6 of A285425.

%Y Cf. A050999, A051000, A051001, A051002, A321811 - A321816 (analog for 2nd .. 12th powers).

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

%Y Cf. A000265, A013665, A013954.

%K nonn,mult

%O 1,3

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