A005063 Sum of squares of primes dividing n.

0, 4, 9, 4, 25, 13, 49, 4, 9, 29, 121, 13, 169, 53, 34, 4, 289, 13, 361, 29, 58, 125, 529, 13, 25, 173, 9, 53, 841, 38, 961, 4, 130, 293, 74, 13, 1369, 365, 178, 29, 1681, 62, 1849, 125, 34, 533, 2209, 13, 49, 29, 298, 173, 2809, 13, 146, 53, 370, 845, 3481, 38, 3721

Offset

1,2

Comments

The set of these terms apart from 0 is A048261. - Bernard Schott, Feb 10 2022

Inverse Möbius transform of n^2 * c(n), where c(n) is the prime characteristic (A010051). - Wesley Ivan Hurt, Jun 22 2024

Links

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Formula

Additive with a(p^e) = p^2.

G.f.: Sum_{k>=1} prime(k)^2*x^prime(k)/(1 - x^prime(k)). - Ilya Gutkovskiy, Dec 24 2016

From Antti Karttunen, Jul 11 2017: (Start)

a(n) = A005066(n) + 4*A059841(n).

a(n) = A005079(n) + A005083(n) + 4*A059841(n).

a(n) = A005071(n) + A005075(n) + 9*A079978(n).

(End)

Dirichlet g.f.: primezeta(s-2)*zeta(s). - Benedict W. J. Irwin, Jul 11 2018

a(n) = Sum_{p|n, p prime} p^2. - Wesley Ivan Hurt, Feb 04 2022

a(n) = Sum_{d|n} d^2 * c(d), where c = A010051. - Wesley Ivan Hurt, Jun 22 2024

Maple

A005063 := proc(n)
        add(d^2, d= numtheory[factorset](n)) ;
end proc;
seq(A005063(n), n=1..40) ; # R. J. Mathar, Nov 08 2011

Mathematica

a[n_] := Total[FactorInteger[n][[All, 1]]^2]; a[1]=0; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Mar 20 2017 *)
Array[DivisorSum[#, #^2 &, PrimeQ] &, 61] (* Michael De Vlieger, Jul 11 2017 *)

Prog

(PARI) a(n)=local(fm, t); fm=factor(n); t=0; for(k=1, matsize(fm)[1], t+=fm[k, 1]^2); t \\ Franklin T. Adams-Watters, May 03 2009
(PARI) a(n) = vecsum(apply(x->x^2, factor(n)[, 1])); \\ Michel Marcus, Sep 19 2020
(Scheme) (define (A005063 n) (if (= 1 n) 0 (+ (A000290 (A020639 n)) (A005063 (A028234 n))))) ;; Antti Karttunen, Jul 10 2017
(Python)
from sympy import primefactors
def a(n): return sum(p**2 for p in primefactors(n))
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jul 11 2017

Crossrefs

Cf. A000290, A005066, A005071, A005075, A005079, A005083, A059841, A079978.

Cf. A067666, A081403, A048261. - Franklin T. Adams-Watters, May 03 2009

Sum of the k-th powers of the primes dividing n for k=0..10 : A001221 (k=0), A008472 (k=1), this sequence (k=2), A005064 (k=3), A005065 (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: A210966 A300516 A178147 * A235323 A345304 A078615

Adjacent sequences: A005060 A005061 A005062 * A005064 A005065 A005066

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Franklin T. Adams-Watters, May 03 2009

Status

approved