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A351249 a(n) = n^9 * Sum_{p|n, p prime} 1/p^9.

%I A351249 #19 Jul 15 2025 22:38:40
%S A351249 0,1,1,512,1,20195,1,262144,19683,1953637,1,10339840,1,40354119,
%T A351249 1972808,134217728,1,397498185,1,1000262144,40373290,2357948203,1,
%U A351249 5293998080,1953125,10604499885,387420489,20661308928,1,39453437071,1,68719476736,2357967374,118587877009,42306732
%N A351249 a(n) = n^9 * Sum_{p|n, p prime} 1/p^9.
%C A351249 Dirichlet convolution of A010051(n) and n^9. - _Wesley Ivan Hurt_, Jul 15 2025
%F A351249 a(A000040(n)) = 1.
%F A351249 a(n) = Sum_{d|n} A069094(d)*A001221(n/d). - _Ridouane Oudra_, Jul 15 2025
%F A351249 From _Wesley Ivan Hurt_, Jul 15 2025: (Start)
%F A351249 a(n) = Sum_{d|n} c(d) * (n/d)^9, where c = A010051.
%F A351249 a(p^k) = p^(9*k-9) for p prime and k>=1. (End)
%e A351249 a(6) = 20195; a(6) = 6^9 * Sum_{p|6, p prime} 1/p^9 = 10077696 * (1/2^9 + 1/3^9) = 20195.
%t A351249 Array[#^9*DivisorSum[#, 1/#^9 &, PrimeQ] &, 50] (* _Wesley Ivan Hurt_, Jul 15 2025 *)
%o A351249 (Python)
%o A351249 from sympy import primefactors
%o A351249 def A351249(n): return sum((n//p)**9 for p in primefactors(n)) # _Chai Wah Wu_, Feb 05 2022
%Y A351249 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), A351248 (k=8), this sequence (k=9), A351262 (k=10).
%Y A351249 Cf. A000040, A001221, A010051, A069094.
%K A351249 nonn
%O A351249 1,4
%A A351249 _Wesley Ivan Hurt_, Feb 05 2022