A254340 Sum of the distinct prime factors of n plus n+1: a(n) = A008472(n) + n + 1.

2, 5, 7, 7, 11, 12, 15, 11, 13, 18, 23, 18, 27, 24, 24, 19, 35, 24, 39, 28, 32, 36, 47, 30, 31, 42, 31, 38, 59, 41, 63, 35, 48, 54, 48, 42, 75, 60, 56, 48, 83, 55, 87, 58, 54, 72, 95, 54, 57, 58, 72, 68, 107, 60, 72, 66, 80, 90, 119, 71, 123, 96, 74, 67, 84

Offset: 1

Author

Wesley Ivan Hurt, May 03 2015

Keywords

• nonn
• easy

Comments

If n is prime, then a(n) = 2n+1; thus if n is a Sophie Germain prime p, then a(p) gives the safe prime q=2p+1.

If n is semiprime, then a(n) = sigma(n).

If m and n are coprime, then a(m*n) = a(m) + a(n) + (m-1)*(n-1) - 2. - Robert Israel, May 04 2015

Crossrefs

Cf. A000203 (sigma), A008472 (sopf), A074372, A075653.

Cf. A005384 (Sophie Germain primes), A005385 (safe primes).

Programs

Magma

[&+PrimeDivisors(n)+n+1: n in [1..70]]; // Bruno Berselli, May 27 2015

Maple

map(t -> t+1+convert(numtheory:-factorset(t),`+`),[$1..100]); # Robert Israel, May 04 2015

Mathematica

Table[n + 1 + DivisorSum[n, # &, PrimeQ[#] &], {n, 100}]

PARI

vector(100,n,vecsum(factor(n)[,1]~)+n+1) \\ Derek Orr, May 13 2015

Formula

a(n) = A075653(n) + 1 = A074372(n) + n. [Bruno Berselli, May 27 2015]