A064372 -id:A064372 - cp's OEIS frontend

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

Ferdinand Rönngren and Lars Kevin Haagensen Strömberg, Aug 22 2020

nonn

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

			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.

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

Cf. A000040, A001414, A064372, A279513.

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

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 *)

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

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.

A287620 a(n) = product, with multiplicity, of the prime numbers appearing at leaf positions in the prime tower factorization of n.

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 3, 2, 10, 11, 6, 13, 14, 15, 2, 17, 4, 19, 10, 21, 22, 23, 9, 2, 26, 3, 14, 29, 30, 31, 5, 33, 34, 35, 4, 37, 38, 39, 15, 41, 42, 43, 22, 10, 46, 47, 6, 2, 4, 51, 26, 53, 6, 55, 21, 57, 58, 59, 30, 61, 62, 14, 6, 65, 66, 67, 34, 69, 70, 71

Offset: 1

Rémy Sigrist, May 28 2017

The prime tower factorization of a number is defined in A182318.
a(n) <= n.
a(n) = n iff n is squarefree (A005117).
a(n) is noncomposite iff n belongs to A164336.
This sequence is surjective; see A287621 for the least k such that a(k) = n.
For n>1, A001222(a(n)) = A064372(n).

			See illustration of the first terms in Links section.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Illustration of the first terms

Cf. A001222, A005117, A064372, A164336, A182318, A287621.

a(n) = my (f=factor(n)); return (prod(i=1, #f~, if (f[i,2]==1, f[i,1], a(f[i,2]))))

Multiplicative with:
- a(p) = p for any prime p,
- a(p^k) = a(k) for any prime p and k > 1.

cp's OEIS Frontend

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

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