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Sum of 10th powers of odd divisors of n.
4

%I #24 Nov 02 2022 10:28:48

%S 1,1,59050,1,9765626,59050,282475250,1,3486843451,9765626,25937424602,

%T 59050,137858491850,282475250,576660215300,1,2015993900450,3486843451,

%U 6131066257802,9765626,16680163512500,25937424602,41426511213650,59050

%N Sum of 10th powers of odd divisors of n.

%H Seiichi Manyama, <a href="/A321814/b321814.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) = A013958(A000265(n)) = sigma_10(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)^10*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^(10*e+10)-1)/(p^10-1) for p > 2.

%F Sum_{k=1..n} a(k) ~ c * n^11, where c = zeta(11)/22 = 0.045477... . (End)

%t a[n_] := DivisorSum[n, #^10 &, OddQ[#] &]; Array[a, 20] (* _Amiram Eldar_, Dec 07 2018 *)

%o (PARI) apply( A321814(n)=sigma(n>>valuation(n,2),10), [1..30]) \\ _M. F. Hasler_, Nov 26 2018

%o (Python)

%o from sympy import divisor_sigma

%o def A321814(n): return int(divisor_sigma(n>>(~n&n-1).bit_length(),10)) # _Chai Wah Wu_, Jul 16 2022

%Y Column k=10 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, A013669, A013958.

%K nonn,mult

%O 1,3

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