A081403 a(n) = A008475(n^2).

0, 4, 9, 16, 25, 13, 49, 64, 81, 29, 121, 25, 169, 53, 34, 256, 289, 85, 361, 41, 58, 125, 529, 73, 625, 173, 729, 65, 841, 38, 961, 1024, 130, 293, 74, 97, 1369, 365, 178, 89, 1681, 62, 1849, 137, 106, 533, 2209, 265, 2401, 629, 298, 185, 2809, 733, 146, 113

Offset: 1

Author

Labos Elemer, Mar 31 2003

Keywords

• nonn

Examples

			a(1) = 0 since 1 has no prime factor; n = p^2: a(p^2) = p^2; n = 6: a(6) = 4+9 = 13; a(u*w) = a(u)+a(w) if gcd(u,w) = 1; a(21) = a(7)+a(3) = 49+9 = 58; additive with respect of unitary prime divisor decompositions.
		

Links

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

Crossrefs

Cf. A008475, A000142, A081403, A081404.

Programs

Maple

a:= n-> add(i[1]^i[2], i=ifactors(n^2)[2]):
seq(a(n), n=1..60);  # Alois P. Heinz, Sep 03 2019

Mathematica

ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] ep[x_] := Table[Part[ffi[x], 2*w], {w, 1, lf[x]}] supo[x_] := Apply[Plus, ba[x]^ep[x]] Table[supo[w], {w, 1, 25}]

PARI

f(n) = { my(f=factor(n)); vecsum(vector(#f~, i, f[i, 1]^f[i, 2])); }; \\ A008475
a(n) = f(n^2); \\ Michel Marcus, Sep 03 2019

Python

from sympy import factorint
def A081403(n): return sum(p**(e<<1) for p,e in factorint(n).items()) # Chai Wah Wu, Jul 01 2024

Formula

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