A068318 Sum of prime factors of n-th semiprime.

4, 5, 6, 7, 9, 8, 10, 13, 10, 15, 14, 19, 12, 21, 16, 25, 14, 20, 16, 22, 31, 33, 18, 26, 39, 18, 43, 22, 45, 32, 20, 34, 49, 24, 55, 40, 28, 61, 24, 22, 63, 44, 46, 26, 69, 50, 73, 24, 34, 75, 36, 81, 56, 30, 85, 26, 62, 91, 64, 42, 28, 99, 70, 103, 36, 46, 105, 30, 74, 109

Offset: 1

Author

Reinhard Zumkeller, Feb 27 2002

Keywords

• nonn

Comments

Odd k is a term if and only if k - 2 is prime. Goldbach's conjecture implies that every even number k >= 4 is a term. - Jianing Song, May 26 2021

Examples

			a(2) = 5 because A001358(2) = 6 = 2*3 and 2+3 = 5.
		

Links

• T. D. Noe, Table of n, a(n) for n=1..1000
• Robert G. Wilson v, Graph of n and a(n).

Crossrefs

Semiprimes are in A001358.

Cf. A084126, A084127, A120831, A120832, A120833, A120834, A109313.

Programs

Maple

with(numtheory): a:=proc(n) if bigomega(n)=2 and nops(factorset(n))=2 then factorset(n)[1]+factorset(n)[2] elif bigomega(n)=2 then 2*sqrt(n) else fi end: seq(a(n),n=1..214); # Emeric Deutsch

Mathematica

f[n_] := Total[#1*#2 & @@@ FactorInteger@ n]; f@# & /@ Select[Range@300, PrimeOmega@# == 2 &] (* Robert G. Wilson v, Jan 23 2013 *)

PARI

s(n) = my(f = factor(n)); if(bigomega(f) == 2, f[,1]~*f[,2], 0);
list(lim) = select(x -> x > 0, apply(s, vector(lim, i, i))); \\ Amiram Eldar, May 15 2025

Python

from math import isqrt
from sympy import primepi, primerange, factorint
def A068318(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return int(n+x+((t:=primepi(s:=isqrt(x)))*(t-1)>>1)-sum(primepi(x//p) for p in primerange(s+1)))
    return sum(p*e for p,e in factorint(bisection(f,n,n)).items()) # Chai Wah Wu, Apr 03 2025

Formula

a(n) = A001414(A001358(n)).

a(n) = A003415(A001358(n)), the arithmetic derivative.

If A001358(n) = s*p, then in this sequence a(n) = s+p.

a(n) = A084126(n)+A084127(n). - Reinhard Zumkeller, Jul 24 2006 [Typo in formula fixed by Zak Seidov, Aug 23 2014]