A345740 a(n) is the least prime p such that Omega(p + n) = n where Omega is A001222, or 0 if no such prime exists.

2, 2, 5, 131, 43, 15619, 281, 6553, 503, 137771, 3061, 244140613, 8179, 22143361, 401393, 199290359, 491503, 8392333984357, 524269, 3486784381, 2097131, 226640986043, 28311529, 303745269775390601, 113246183, 9885033776809, 469762021, 176518460300597, 805306339, 77737724676061053405079339

Offset: 1

Author

Zak Seidov, Jun 25 2021

Keywords

• nonn

Examples

			For n=1, a(1) = 2 as 2+1 = 3 (Omega(2 + 1) = Omega(3) = 1, see A000040(1)).
For n=2, 2+2 = 4 = 2*2 (semiprime, Omega(4) = 2, see A001358(1)).
For n=3, 5+3 = 8 = 2*2*2 (triprime, Omega(8) = 3, see A014612(1)).
For n=4, 131+4 = 135 = 3*3*3*5 (Omega(135) = 4,  see A014613(16)).
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..1049

Crossrefs

Cf. A000040, A001222, A001358, A014612, A014613, A053669, A069279, A097977.

Programs

Mathematica

Table[k=1;While[PrimeOmega[Prime@k+n]!=n,k++];Prime@k,{n,11}] (* Giorgos Kalogeropoulos, Jun 25 2021 *)

PARI

a(n) = my(p=2); while (bigomega(p+n) != n, p = nextprime(p+1)); p; \\ Michel Marcus, Jun 26 2021

Python

from sympy import factorint, nextprime, primerange
def Omega(n): return sum(e for f, e in factorint(n).items())
def a(n):
    lb = 2**n
    p = nextprime(max(lb-n, 1) - 1)
    while Omega(p+n) != n: p = nextprime(p)
    return p
print([a(n) for n in range(1, 12)]) # Michael S. Branicky, Aug 14 2021

Formula

a(n) + n >= A053669(n)^n for n > 2 if a(n) exists. - David A. Corneth, Aug 14 2021