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A351262 a(n) = n^10 * Sum_{p|n, p prime} 1/p^10.

%I A351262 #25 Jul 15 2025 22:36:58
%S A351262 0,1,1,1024,1,60073,1,1048576,59049,9766649,1,61514752,1,282476273,
%T A351262 9824674,1073741824,1,3547250577,1,10001048576,282534298,25937425625,
%U A351262 1,62991106048,9765625,137858492873,3486784401,289255703552,1,586710856801,1,1099511627776,25937483650
%N A351262 a(n) = n^10 * Sum_{p|n, p prime} 1/p^10.
%C A351262 Dirichlet convolution of A010051(n) and n^10. - _Wesley Ivan Hurt_, Jul 15 2025
%H A351262 Robert Israel, <a href="/A351262/b351262.txt">Table of n, a(n) for n = 1..10000</a>
%F A351262 a(A000040(n)) = 1.
%F A351262 a(n) = Sum_{d|n} A069095(d)*A001221(n/d). - _Ridouane Oudra_, Jul 15 2025
%F A351262 From _Wesley Ivan Hurt_, Jul 15 2025: (Start)
%F A351262 a(n) = Sum_{d|n} c(d) * (n/d)^10, where c = A010051.
%F A351262 a(p^k) = p^(10*k-10) for p prime and k>=1. (End)
%e A351262 a(6) = 60073; a(6) = 6^10 * Sum_{p|6, p prime} 1/p^10 = 60466176 * (1/2^10 + 1/3^10) = 60073.
%p A351262 f:= proc(n) local p;
%p A351262   n^10 * add(1/p^10, p = numtheory:-factorset(n))
%p A351262 end proc:
%p A351262 map(f, [$1..40]); # _Robert Israel_, Sep 10 2024
%t A351262 Join[{0},Table[n^10 Total[1/FactorInteger[n][[;;,1]]^10],{n,2,40}]] (* _Harvey P. Dale_, Aug 10 2024 *)
%o A351262 (Python)
%o A351262 from sympy import primefactors
%o A351262 def A351262(n): return sum((n//p)**10 for p in primefactors(n)) # _Chai Wah Wu_, Feb 05 2022
%o A351262 (PARI) a(n) = my(f=factor(n)); n^10*sum(k=1, #f~, 1/f[k,1]^10); \\ _Michel Marcus_, Sep 10 2024
%Y A351262 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), A351249 (k=9), this sequence (k=10).
%Y A351262 Cf. A000040, A001221, A010051, A069095.
%K A351262 nonn
%O A351262 1,4
%A A351262 _Wesley Ivan Hurt_, Feb 05 2022