A337310 Additive function with a(p) = p, a(p^e) = p*a(e) for prime p and e > 1, with a(1) = 1.

1, 2, 3, 4, 5, 5, 7, 6, 6, 7, 11, 7, 13, 9, 8, 8, 17, 8, 19, 9, 10, 13, 23, 9, 10, 15, 9, 11, 29, 10, 31, 10, 14, 19, 12, 10, 37, 21, 16, 11, 41, 12, 43, 15, 11, 25, 47, 11, 14, 12, 20, 17, 53, 11, 16, 13, 22, 31, 59, 12, 61, 33, 13, 10, 18, 16, 67, 21, 26, 14, 71, 12, 73

Offset: 1

Author

Ferdinand Rönngren and Lars Kevin Haagensen Strömberg, Aug 22 2020

Keywords

• nonn

Comments

a(n) <= A001414(n) for n > 1, with equality if and only if all the exponents in the prime factorization of n are either less than 6 or prime themselves. - Mital Ashok, Jun 22 2025

Examples

			a(100) = a(2^2*5^2) = 2*a(2) + 5*a(2) = 2*2 + 5*2 = 14.
a(192) = a(2^6*3^1) = 2*a(6) + 3*a(1) = 2*a(2^1*3^1) + 3*1 = 2*(2*a(1) + 3*a(1)) + 3 = 2*(2*1 + 3*1) + 3 = 13.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

Cf. A000040, A001414, A064372, A279513.

Programs

Maple

a:= proc(n) option remember; `if`(n=1, 1,
      add(i[1]*a(i[2]), i=ifactors(n)[2]))
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Aug 22 2020

Mathematica

f[p_, e_] := p * a[e]; a[1] = 1; a[n_] := a[n] = Plus @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Aug 22 2020 *)

PARI

a(n)={my(f=factor(n)); if(n==1, 1, sum(i=1, #f~, my([p,e]=f[i,]); p*a(e)))} \\ Andrew Howroyd, Aug 22 2020

Formula

a(1)=1, a(p_1^b_1*p_2^b_2*...*p_n^b_n)=p_1*a(b_1)+p_2*a(b_2)+...+p_n*a(b_n) where p_i is the i-th prime number.