A005065 Sum of 4th powers of primes dividing n.

0, 16, 81, 16, 625, 97, 2401, 16, 81, 641, 14641, 97, 28561, 2417, 706, 16, 83521, 97, 130321, 641, 2482, 14657, 279841, 97, 625, 28577, 81, 2417, 707281, 722, 923521, 16, 14722, 83537, 3026, 97, 1874161, 130337, 28642, 641, 2825761, 2498, 3418801, 14657, 706, 279857, 4879681, 97, 2401, 641, 83602, 28577, 7890481, 97

Offset

1,2

Comments

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

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

Links

Antti Karttunen, Table of n, a(n) for n = 1..10000

Formula

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

From Antti Karttunen, Jul 11 2017: (Start)

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

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

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

(End)

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

a(n) = Sum_{p|n, p prime} p^4. - Wesley Ivan Hurt, Feb 04 2022

a(n) = Sum_{d|n} d^4 * c(d), where c = A010051. - Wesley Ivan Hurt, Jun 22 2024

Maple

A005065 := proc(n)
        add(d^4, d= numtheory[factorset](n)) ;
end proc;
seq(A005065(n), n=1..40) ; # R. J. Mathar, Nov 08 2011

Mathematica

Join[{0}, Table[Total[Transpose[FactorInteger[n]][[1]]^4], {n, 2, 40}]] (* Harvey P. Dale, Jul 16 2014 *)
Array[DivisorSum[#, #^4 &, PrimeQ] &, 54] (* Michael De Vlieger, Jul 11 2017 *)

Prog

(Scheme) (define (A005065 n) (if (= 1 n) 0 (+ (A000583 (A020639 n)) (A005065 (A028234 n))))) ;; Antti Karttunen, Jul 10 2017
(Python)
from sympy import primefactors
def a(n): return sum(p**4 for p in primefactors(n))
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jul 11 2017
(PARI) a(n) = my(f=factor(n)); sum(k=1, #f~, f[k, 1]^4); \\ Michel Marcus, Jul 11 2017

Crossrefs

Column k=4 of A322080.

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

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).

Cf. A010051.

Sequence in context: A330480 A334660 A088379 * A056553 A053167 A187363

Adjacent sequences: A005062 A005063 A005064 * A005066 A005067 A005068

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Antti Karttunen, Jul 10 2017

Status

approved