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a(n) = n^7 * Sum_{d^2|n} 1 / d^7.
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%I #30 Nov 14 2022 01:39:45

%S 1,128,2187,16512,78125,279936,823543,2113536,4785156,10000000,

%T 19487171,36111744,62748517,105413504,170859375,270548992,410338673,

%U 612499968,893871739,1290000000,1801088541,2494357888,3404825447,4622303232,6103593750,8031810176,10465136172

%N a(n) = n^7 * Sum_{d^2|n} 1 / d^7.

%H Seiichi Manyama, <a href="/A351605/b351605.txt">Table of n, a(n) for n = 1..10000</a>

%F Multiplicative with a(p^e) = p^7*(p^(7*e) - p^(7*floor((e-1)/2)))/(p^7 - 1). - _Sebastian Karlsson_, Feb 25 2022

%F Sum_{k=1..n} a(k) ~ c * n^8, where c = zeta(9)/8 = 0.125251... . - _Amiram Eldar_, Nov 13 2022

%t f[p_, e_] := p^7*(p^(7*e) - p^(7*Floor[(e - 1)/2]))/(p^7 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 30] (* _Amiram Eldar_, Nov 13 2022 *)

%o (PARI) a(n) = n^7*sumdiv(n, d, if (issquare(d), 1/sqrtint(d^7))); \\ _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), this sequence (k=7), A351606 (k=8), A351607 (k=9), A351608 (k=10).

%Y Cf. A013667.

%K nonn,mult

%O 1,2

%A _Wesley Ivan Hurt_, Feb 14 2022