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A351245 a(n) = n^5 * Sum_{p|n, p prime} 1/p^5.

%I A351245 #38 Jul 15 2025 22:44:47
%S A351245 0,1,1,32,1,275,1,1024,243,3157,1,8800,1,16839,3368,32768,1,66825,1,
%T A351245 101024,17050,161083,1,281600,3125,371325,59049,538848,1,867151,1,
%U A351245 1048576,161294,1419889,19932,2138400,1,2476131,371536,3232768,1,4629701,1,5154656,818424,6436375,1
%N A351245 a(n) = n^5 * Sum_{p|n, p prime} 1/p^5.
%C A351245 Dirichlet convolution of A010051(n) and n^5. - _Wesley Ivan Hurt_, Jul 15 2025
%H A351245 Seiichi Manyama, <a href="/A351245/b351245.txt">Table of n, a(n) for n = 1..10000</a>
%F A351245 a(A000040(n)) = 1.
%F A351245 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 A351245 Sum_{k=1..n} a(k) ~ A085966 * n^6/6. - _Vaclav Kotesovec_, Mar 03 2023
%F A351245 a(n) = Sum_{d|n} A059378(d)*A001221(n/d). - _Ridouane Oudra_, Jul 14 2025
%F A351245 From _Wesley Ivan Hurt_, Jul 15 2025: (Start)
%F A351245 a(n) = Sum_{d|n} c(d) * (n/d)^5, where c = A010051.
%F A351245 a(p^k) = p^(5*k-5) for p prime and k>=1. (End)
%e A351245 a(6) = 275; a(6) = 6^5 * Sum_{p|6, p prime} 1/p^5 = 7776 * (1/2^5 + 1/3^5) = 275.
%t A351245 Array[#^5*DivisorSum[#, 1/#^5 &, PrimeQ] &, 47] (* _Stefano Spezia_, Jul 15 2025 *)
%o A351245 (PARI) a(n) = my(f = factor(n)); sum(i = 1, #f~, (n/f[i,1])^5) \\ _David A. Corneth_, Jul 15 2025
%Y A351245 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 A351245 Cf. A000040, A001221, A010051, A059378, A085966.
%K A351245 nonn
%O A351245 1,4
%A A351245 _Wesley Ivan Hurt_, Feb 05 2022