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

1,1

Links

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

Formula

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

Example

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)).

Mathematica

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

Prog

(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

Crossrefs

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

Sequence in context: A114715 A270750 A234703 * A230607 A353534 A036739

Adjacent sequences: A345737 A345738 A345739 * A345741 A345742 A345743

Keyword

nonn

Author

Zak Seidov, Jun 25 2021

Status

approved