A075653 a(n) = n + sopf(n), where sopf is the sum of the distinct prime factors of n (A008472).

1, 4, 6, 6, 10, 11, 14, 10, 12, 17, 22, 17, 26, 23, 23, 18, 34, 23, 38, 27, 31, 35, 46, 29, 30, 41, 30, 37, 58, 40, 62, 34, 47, 53, 47, 41, 74, 59, 55, 47, 82, 54, 86, 57, 53, 71, 94, 53, 56, 57, 71, 67, 106, 59, 71, 65, 79, 89, 118, 70, 122, 95, 73, 66, 83, 82, 134, 87, 95

Offset: 1

Author

Joseph L. Pe, Oct 11 2002

Keywords

• easy
• nonn

Comments

For 1 <= k <= n, add sigma(k) if k is a prime factor of n, otherwise add 1. For example, a(6) = 11 since for k = 1,2,.. we have 1 + sigma(2) + sigma(3) + 1 + 1 + 1 = 1 + (1+2) + (1+3) + 1 + 1 + 1 = 11. - Wesley Ivan Hurt, Oct 18 2021

Examples

			6 + sum of prime factors of 6 = 6 + 2 + 3 = 11, so a(6) = 11.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A008472, A075254, A075255, A076694.

Programs

Mathematica

Flatten[Append[{1}, Table[n + Apply[Plus, Transpose[FactorInteger[n]][[1]]], {n, 2, 100}]]]
Join[{1},Table[n+Total[FactorInteger[n][[All,1]]],{n,2,70}]] (* Harvey P. Dale, Sep 29 2016 *)

PARI

a(n) = n + vecsum(factor(n)[,1]); \\ Michel Marcus, Feb 22 2017

Formula

a(n) = n + A008472(n).

a(p) = 2p for primes p. - Wesley Ivan Hurt, Oct 18 2021