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Sum of the 5th powers of primes dividing n.
11

%I #21 Jun 22 2024 18:29:15

%S 0,32,243,32,3125,275,16807,32,243,3157,161051,275,371293,16839,3368,

%T 32,1419857,275,2476099,3157,17050,161083,6436343,275,3125,371325,243,

%U 16839,20511149,3400,28629151,32,161294,1419889,19932,275,69343957,2476131,371536,3157

%N Sum of the 5th powers of primes dividing n.

%C Inverse Möbius transform of n^5 * c(n), where c(n) is the prime characteristic (A010051). - _Wesley Ivan Hurt_, Jun 22 2024

%H Robert Israel, <a href="/A351193/b351193.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = Sum_{p|n, p prime} p^5.

%F G.f.: Sum_{k>=1} prime(k)^5 * x^prime(k) / (1 - x^prime(k)). - _Ilya Gutkovskiy_, Feb 16 2022

%F Additive with a(p^e) = p^5. - _Amiram Eldar_, Jun 20 2022

%F a(n) = Sum_{d|n} d^5 * c(d), where c = A010051. - _Wesley Ivan Hurt_, Jun 22 2024

%p f:= n -> add(p^5, p = numtheory:-factorset(n)):

%p map(f, [$1..100]); # _Robert Israel_, Feb 18 2022

%t Array[DivisorSum[#, #^5 &, PrimeQ] &, 50]

%t f[p_, e_] := p^5; a[n_] := Plus @@ f @@@ FactorInteger[n]; a[1] = 0; Array[a, 100] (* _Amiram Eldar_, Jun 20 2022 *)

%Y Sum of the k-th powers of the primes dividing n for k=0..10 : A001221 (k=0), A008472 (k=1), A005063 (k=2), A005064 (k=3), A005065 (k=4), this sequence (k=5), A351194 (k=6), A351195 (k=7), A351196 (k=8), A351197 (k=9), A351198 (k=10).

%Y Cf. A010051.

%K nonn

%O 1,2

%A _Wesley Ivan Hurt_, Feb 04 2022