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

1,2

Links

Table of n, a(n) for n=1..73.

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.

Example

a(100) = a(2^2*5^2) = 2*a(2) + 5*a(2) = 2*2*a(1) + 5*2*a(1) = 2*2*1 + 5*2*1 = 14.

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 *)

Prog

(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

Crossrefs

Cf. A000040, A001414, A064372, A279513.

Sequence in context: A345423 A159303 A262049 * A001414 A134875 A134889

Adjacent sequences: A337307 A337308 A337309 * A337311 A337312 A337313

Keyword

nonn

Author

Ferdinand Rönngren and Lars Kevin Haagensen Strömberg, Aug 22 2020

Status

approved