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a(n) = n^9 * Sum_{d^2|n} 1 / d^9.
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%I #29 Nov 14 2022 01:39:54

%S 1,512,19683,262656,1953125,10077696,40353607,134479872,387440172,

%T 1000000000,2357947691,5169858048,10604499373,20661046784,38443359375,

%U 68853956608,118587876497,198369368064,322687697779,513000000000,794280046581,1207269217792,1801152661463

%N a(n) = n^9 * Sum_{d^2|n} 1 / d^9.

%H Seiichi Manyama, <a href="/A351607/b351607.txt">Table of n, a(n) for n = 1..10000</a>

%F Multiplicative with a(p^e) = p^9*(p^(9*e) - p^(9*floor((e-1)/2)))/(p^9 - 1). - _Sebastian Karlsson_, Mar 03 2022

%F Sum_{k=1..n} a(k) ~ c * n^10, where c = zeta(11)/10 = 0.100049... . - _Amiram Eldar_, Nov 13 2022

%t f[p_, e_] := p^9*(p^(9*e) - p^(9*Floor[(e - 1)/2]))/(p^9 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 30] (* _Amiram Eldar_, Nov 13 2022 *)

%o (PARI) a(n) = n^9*sumdiv(n, d, if (issquare(d), 1/sqrtint(d^9))); \\ _Michel Marcus_, Feb 15 2022

%Y Sequences of the form n^k * Sum_{d^2|n} 1/d^k for k = 0..10: A046951 (k=0), A340774 (k=1), A351600 (k=2), A351601 (k=3), A351602 (k=4), A351603 (k=5), A351604 (k=6), A351605 (k=7), A351606 (k=8), this sequence (k=9), A351608 (k=10).

%Y Cf. A013669.

%K nonn,mult

%O 1,2

%A _Wesley Ivan Hurt_, Feb 14 2022