A335403 If n = Product_{i=1..k} p_i^e_i then a(n) = Sum_{i=1..k} e_i * prime(p_i).

0, 3, 5, 6, 11, 8, 17, 9, 10, 14, 31, 11, 41, 20, 16, 12, 59, 13, 67, 17, 22, 34, 83, 14, 22, 44, 15, 23, 109, 19, 127, 15, 36, 62, 28, 16, 157, 70, 46, 20, 179, 25, 191, 37, 21, 86, 211, 17, 34, 25, 64, 47, 241, 18, 42, 26, 72, 112, 277, 22, 283, 130, 27, 18

Offset: 1

Author

Gus Wiseman, Jun 06 2020

Keywords

• nonn
• look

Comments

Totally additive with a(p) = prime(p) for p prime.

Examples

			The prime factors of 18 are 2 * 3 * 3, so a(18) = prime(2) + prime(3) + prime(3) = 13.
		

Links

• Wikipedia, Additive function

Crossrefs

Partitions into prime parts are A000607.

Sum of prime factors is A001414.

Primes of prime index are A006450.

Sum of prime indices is A056239.

The multiplicative version is A064988.

Products of primes of prime index are A076610.

Numbers whose prime indices are not all prime are A330945.

Cf. A000040, A000720, A001222, A003961, A003963, A007097, A112798, A178503, A257994, A302242, A324851, A331915.

Programs

Mathematica

Table[Total[Cases[FactorInteger[n],{p_,k_}:>k*Prime[p]]],{n,30}]

PARI

a(n) = my(f=factor(n)); sum(k=1, #f~, prime(f[k,1])*f[k,2]); \\ Michel Marcus, Jun 07 2020

Extensions

Edited by N. J. A. Sloane, Jun 20 2020 following a suggestion from Bernard Schott.