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

Author

N. J. A. Sloane

Keywords

• nonn

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

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.

Programs

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 *)

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

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

Scheme

(define (A005063 n) (if (= 1 n) 0 (+ (A000290 (A020639 n)) (A005063 (A028234 n))))) ;; Antti Karttunen, Jul 10 2017

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

Extensions

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