A385997 a(n) is the smallest k such that the sum of the first k primes has exactly n prime factors, counting multiplicity.

1, 3, 5, 9, 17, 11, 103, 119, 475, 237, 2661, 1481, 3045, 1567, 22019, 34907, 24995, 28173, 4915, 269225, 214183, 927571, 1085315, 9724983, 2567053, 4620383, 8827803, 38175467, 37167809, 98773463, 153124063, 257222427, 370283099, 24322477, 592786617

Offset: 1

Author

Felix Huber, Jul 27 2025

Keywords

• nonn
• more
• changed

Comments

a(n) is the smallest k such that A001222(A007504(k)) = A102862(k) = n.

a(34) = 24322477. - Robert Israel, Jul 29 2025

Examples

			a(2) = 3, because the sum of the first three primes 2 + 3 + 5 = 10 = 2*5 has exactly 2 prime factors. The sums of the first 1 or 2 primes (2 or 2 + 3 = 5) have only one prime factor.
a(5) = 17, because the sum of the first 17 primes (440 = 2^3*5*11) has exactly 5 prime factors. The sums of the first 1, 2, ..., 16 primes have either fewer or more than 5 prime factors.
		

Crossrefs

Cf. A000040, A001222, A007504, A102862.

Programs

Maple

M:= 29: # for a(1) .. a(M)
V:= Vector(M):
t:= 0: p:= 1: count:= 0:
for i from 1 while count < M do
  p:= nextprime(p);
  t:= t + p;
  v:= numtheory:-bigomega(t);
  if v <= M and V[v] = 0 then V[v]:= i; count:= count+1 fi
od:
convert(V,list); # Robert Israel, Jul 29 2025

Mathematica

a[n_]:=Module[{k=1,ps=0},Until[PrimeOmega[ps]==n,ps=ps+Prime[k];k++];k-1];Array[a,20] (* James C. McMahon, Aug 05 2025 *)

Python

from itertools import count
from sympy import factorint, nextprime
def A385997(n):
    p, c = 2, 0
    for k in count(1):
        c += p
        if sum(factorint(c).values())==n:
            return k
        p = nextprime(p) # Chai Wah Wu, Aug 08 2025

Extensions

a(23)-a(30) from Robert Israel, Jul 29 2025

a(31) from Sean A. Irvine, Aug 05 2025

a(32)-a(34) from Chai Wah Wu, Aug 08 2025

a(35) from Chai Wah Wu, Sep 01 2025