login
Sum of the 9th powers of the primes dividing n.
11

%I #22 Jun 22 2024 18:42:05

%S 0,512,19683,512,1953125,20195,40353607,512,19683,1953637,2357947691,

%T 20195,10604499373,40354119,1972808,512,118587876497,20195,

%U 322687697779,1953637,40373290,2357948203,1801152661463,20195,1953125,10604499885,19683,40354119,14507145975869,1973320,26439622160671,512,2357967374

%N Sum of the 9th powers of the primes dividing n.

%C Inverse Möbius transform of n^9 * c(n), where c(n) is the prime characteristic (A010051). - _Wesley Ivan Hurt_, Jun 22 2024

%H Seiichi Manyama, <a href="/A351197/b351197.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = Sum_{p|n, p prime} p^9.

%F G.f.: Sum_{k>=1} prime(k)^9 * x^prime(k) / (1 - x^prime(k)). - _Ilya Gutkovskiy_, Feb 16 2022

%F Additive with a(p^e) = p^9. - _Amiram Eldar_, Jun 20 2022

%F a(n) = Sum_{d|n} d^9 * c(d), where c = A010051. - _Wesley Ivan Hurt_, Jun 22 2024

%t Array[DivisorSum[#, #^9 &, PrimeQ] &, 50]

%t f[p_, e_] := p^9; a[n_] := Plus @@ f @@@ FactorInteger[n]; a[1] = 0; Array[a, 100] (* _Amiram Eldar_, Jun 20 2022 *)

%o (Python)

%o from sympy import primefactors

%o def A351197(n): return sum(p**9 for p in primefactors(n)) # _Chai Wah Wu_, Feb 05 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), A351193 (k=5), A351194 (k=6), A351195 (k=7), A351196 (k=8), this sequence (k=9), A351198 (k=10).

%Y Cf. A010051.

%K nonn

%O 1,2

%A _Wesley Ivan Hurt_, Feb 04 2022