A067666 Sum of squares of prime factors of n (counted with multiplicity).

0, 4, 9, 8, 25, 13, 49, 12, 18, 29, 121, 17, 169, 53, 34, 16, 289, 22, 361, 33, 58, 125, 529, 21, 50, 173, 27, 57, 841, 38, 961, 20, 130, 293, 74, 26, 1369, 365, 178, 37, 1681, 62, 1849, 129, 43, 533, 2209, 25, 98, 54, 298, 177, 2809, 31, 146, 61, 370, 845, 3481

Offset: 1

Author

Henry Bottomley, Feb 04 2002

Keywords

• nonn

Comments

16 and 27 are fixed points, ... and see Rivera link. - Michel Marcus, Sep 19 2020

Examples

			a(2) = 2^2 = 4;
a(45) = a(3*3*5) = 3^2 + 3^2 + 5^2 = 43.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• Carlos Rivera, Puzzle 625. Sum of squares of prime divisors, The Prime Puzzles and Problems Connection.
• Carlos Rivera, Puzzle 1019. Follow-up to Puzzle 625, The Prime Puzzles and Problems Connection.

Crossrefs

Cf. A166319 (where a(n)<=n), A001222, A001414, A005063, A078137, A081403.

Programs

Maple

A067666 := proc(n)
    add(op(2,pe)*op(1,pe)^2, pe=ifactors(n)[2]) ;
end proc:
seq(A067666(n),n=1..100) ;# R. J. Mathar, Jul 31 2024

Mathematica

Join[{0},Table[Total[Flatten[Table[#[[1]],{#[[2]]}]&/@ FactorInteger[ n]]^2],{n,2,60}]] (* Harvey P. Dale, Dec 24 2012 *)
Join[{0}, Table[Total[#[[1]]^2*#[[2]] & /@ FactorInteger[n]], {n, 2, 60}]] (* Zak Seidov, Apr 18 2013 *)

PARI

a(n)=local(fm,t);fm=factor(n);t=0;for(k=1,matsize(fm)[1],t+=fm[k,1]^2*fm[k,2]);t \\ Franklin T. Adams-Watters, May 03 2009

PARI

a(n) = my(f=factor(n)); sum(k=1, #f~, f[k,1]^2*f[k,2]); \\ Michel Marcus, Sep 19 2020

Formula

a(x*y) = a(x) + a(y); a(p^k) = k*p^2 for p prime.

Totally additive with a(p) = p^2.

Extensions

Values through a(59) verified by Franklin T. Adams-Watters, May 03 2009