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Sum of 8th powers of odd divisors of n.
4

%I #23 Nov 02 2022 10:28:52

%S 1,1,6562,1,390626,6562,5764802,1,43053283,390626,214358882,6562,

%T 815730722,5764802,2563287812,1,6975757442,43053283,16983563042,

%U 390626,37828630724,214358882,78310985282,6562,152588281251,815730722,282472589764

%N Sum of 8th powers of odd divisors of n.

%H Seiichi Manyama, <a href="/A321812/b321812.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) = A013956(A000265(n)) = sigma_8(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)^8*x^(2*k-1)/(1 - x^(2*k-1)). - _Ilya Gutkovskiy_, Dec 07 2018

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

%F Multiplicative with a(2^e) = 1 and a(p^e) = (p^(8*e+8)-1)/(p^8-1) for p > 2.

%F Sum_{k=1..n} a(k) ~ c * n^9, where c = zeta(9)/18 = 0.0556671... . (End)

%t a[n_] := DivisorSum[n, #^8 &, OddQ[#] &]; Array[a, 20] (* _Amiram Eldar_, Dec 07 2018 *)

%o (PARI) apply( A321812(n)=sigma(n>>valuation(n,2),8), [1..30]) \\ _M. F. Hasler_, Nov 26 2018

%o (Python)

%o from sympy import divisor_sigma

%o def A321812(n): return int(divisor_sigma(n>>(~n&n-1).bit_length(),8)) # _Chai Wah Wu_, Jul 16 2022

%Y Column k=8 of A285425.

%Y Cf. A050999, A051000, A051001, A051002, A321810 - A321816 (analog for 2nd .. 12th powers).

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

%Y Cf. A000265, A013667, A013956.

%K nonn,mult

%O 1,3

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