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a(n) = n^5 * Sum_{p|n, p prime} 1/p^5.
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%I #24 Mar 09 2023 20:50:07

%S 0,1,1,32,1,275,1,1024,243,3157,1,8800,1,16839,3368,32768,1,66825,1,

%T 101024,17050,161083,1,281600,3125,371325,59049,538848,1,867151,1,

%U 1048576,161294,1419889,19932,2138400,1,2476131,371536,3232768,1,4629701,1,5154656,818424,6436375,1

%N a(n) = n^5 * Sum_{p|n, p prime} 1/p^5.

%H Seiichi Manyama, <a href="/A351245/b351245.txt">Table of n, a(n) for n = 1..10000</a>

%F a(A000040(n)) = 1.

%F Dirichlet g.f.: zeta(s-5)*primezeta(s). This follows because Sum_{n>=1} a(n)/n^s = Sum_{n>=1} (n^5/n^s) Sum_{p|n} 1/p^5. Since n = p*j, rewrite the sum as Sum_{p} Sum_{j>=1} 1/(p^5*(p*j)^(s-5)) = Sum_{p} 1/p^s Sum_{j>=1} 1/j^(s-5) = zeta(s-5)*primezeta(s). The result generalizes to higher powers of p. - _Michael Shamos_, Mar 03 2023

%F Sum_{k=1..n} a(k) ~ A085966 * n^6/6. - _Vaclav Kotesovec_, Mar 03 2023

%e a(6) = 275; a(6) = 6^5 * Sum_{p|6, p prime} 1/p^5 = 7776 * (1/2^5 + 1/3^5) = 275.

%Y Sequences of the form n^k * Sum_{p|n, p prime} 1/p^k for k = 0..10: A001221 (k=0), A069359 (k=1), A322078 (k=2), A351242 (k=3), A351244 (k=4), this sequence (k=5), A351246 (k=6), A351247 (k=7), A351248 (k=8), A351249 (k=9), A351262 (k=10).

%Y Cf. A000040, A085966.

%K nonn

%O 1,4

%A _Wesley Ivan Hurt_, Feb 05 2022