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

1,2

Comments

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

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.

Formula

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

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

Example

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

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A285109 A217579 A118585 * A355012 A280286 A268597

Adjacent sequences: A067663 A067664 A067665 * A067667 A067668 A067669

Keyword

nonn

Author

Henry Bottomley, Feb 04 2002

Extensions

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

Status

approved