login
Sum of 4th powers of primes dividing n.
20

%I #36 Jun 22 2024 18:59:46

%S 0,16,81,16,625,97,2401,16,81,641,14641,97,28561,2417,706,16,83521,97,

%T 130321,641,2482,14657,279841,97,625,28577,81,2417,707281,722,923521,

%U 16,14722,83537,3026,97,1874161,130337,28642,641,2825761,2498,3418801,14657,706,279857,4879681,97,2401,641,83602,28577,7890481,97

%N Sum of 4th powers of primes dividing n.

%C Primes are taken without multiplicity, e.g., 12 = 2*2*3, and a(12) = 2^4+3^4 = 97. - _Harvey P. Dale_, Jul 16 2014

%C Inverse Möbius transform of n^4 * c(n), where c(n) is the prime characteristic (A010051). - _Wesley Ivan Hurt_, Jun 22 2024

%H Antti Karttunen, <a href="/A005065/b005065.txt">Table of n, a(n) for n = 1..10000</a>

%F Additive with a(p^e) = p^4.

%F From _Antti Karttunen_, Jul 11 2017: (Start)

%F a(n) = A005068(n) + 16*A059841(n).

%F a(n) = A005081(n) + A005085(n) + 16*A059841(n).

%F a(n) = A005073(n) + A005077(n) + 81*A079978(n).

%F (End)

%F G.f.: Sum_{k>=1} prime(k)^4*x^prime(k)/(1 - x^prime(k)). - _Ilya Gutkovskiy_, Dec 24 2018

%F a(n) = Sum_{p|n, p prime} p^4. - _Wesley Ivan Hurt_, Feb 04 2022

%F a(n) = Sum_{d|n} d^4 * c(d), where c = A010051. - _Wesley Ivan Hurt_, Jun 22 2024

%p A005065 := proc(n)

%p add(d^4, d= numtheory[factorset](n)) ;

%p end proc;

%p seq(A005065(n),n=1..40) ; # _R. J. Mathar_, Nov 08 2011

%t Join[{0},Table[Total[Transpose[FactorInteger[n]][[1]]^4],{n,2,40}]] (* _Harvey P. Dale_, Jul 16 2014 *)

%t Array[DivisorSum[#, #^4 &, PrimeQ] &, 54] (* _Michael De Vlieger_, Jul 11 2017 *)

%o (Scheme) (define (A005065 n) (if (= 1 n) 0 (+ (A000583 (A020639 n)) (A005065 (A028234 n))))) ;; _Antti Karttunen_, Jul 10 2017

%o (Python)

%o from sympy import primefactors

%o def a(n): return sum(p**4 for p in primefactors(n))

%o print([a(n) for n in range(1, 101)]) # _Indranil Ghosh_, Jul 11 2017

%o (PARI) a(n) = my(f=factor(n)); sum(k=1, #f~, f[k,1]^4); \\ _Michel Marcus_, Jul 11 2017

%Y Column k=4 of A322080.

%Y Cf. A000583, A005068, A005073, A005077, A005081, A005085, A008472, A059841, A079978.

%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), this sequence (k=4), A351193 (k=5), A351194 (k=6), A351195 (k=7), this sequence (k=8), A351197 (k=9), A351198 (k=10).

%Y Cf. A010051.

%K nonn

%O 1,2

%A _N. J. A. Sloane_

%E More terms from _Antti Karttunen_, Jul 10 2017