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A351248 a(n) = n^8 * Sum_{p|n, p prime} 1/p^8.

%I A351248 #22 Jul 15 2025 22:39:52
%S A351248 0,1,1,256,1,6817,1,65536,6561,390881,1,1745152,1,5765057,397186,
%T A351248 16777216,1,44726337,1,100065536,5771362,214359137,1,446758912,390625,
%U A351248 815730977,43046721,1475854592,1,2664570241,1,4294967296,214365442,6975757697,6155426,11449942272
%N A351248 a(n) = n^8 * Sum_{p|n, p prime} 1/p^8.
%C A351248 Dirichlet convolution of A010051(n) and n^8. - _Wesley Ivan Hurt_, Jul 15 2025
%F A351248 a(A000040(n)) = 1.
%F A351248 a(n) = Sum_{d|n} A069093(d)*A001221(n/d). - _Ridouane Oudra_, Jul 15 2025
%F A351248 From _Wesley Ivan Hurt_, Jul 15 2025: (Start)
%F A351248 a(n) = Sum_{d|n} c(d) * (n/d)^8, where c = A010051.
%F A351248 a(p^k) = p^(8*k-8) for p prime and k>=1. (End)
%e A351248 a(6) = 6817; a(6) = 6^8 * Sum_{p|6, p prime} 1/p^8 = 1679616 * (1/2^8 + 1/3^8) = 6817.
%t A351248 Array[#^8*DivisorSum[#, 1/#^8 &, PrimeQ] &, 36] (* _Stefano Spezia_, Jul 15 2025 *)
%o A351248 (Python)
%o A351248 from sympy import primefactors
%o A351248 def A351248(n): return sum((n//p)**8 for p in primefactors(n)) # _Chai Wah Wu_, Feb 05 2022
%Y A351248 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), A351245 (k=5), A351246 (k=6), A351247 (k=7), this sequence (k=8), A351249 (k=9), A351262 (k=10).
%Y A351248 Cf. A000040, A001221, A010051, A069093.
%K A351248 nonn
%O A351248 1,4
%A A351248 _Wesley Ivan Hurt_, Feb 05 2022