A082130 Numbers k such that 2*k-1 and 2*k+1 are semiprimes.

17, 25, 28, 43, 46, 47, 60, 61, 71, 72, 80, 92, 93, 101, 102, 107, 108, 109, 110, 118, 124, 133, 144, 145, 150, 151, 152, 160, 161, 164, 170, 196, 197, 206, 207, 208, 223, 226, 235, 236, 258, 259, 264, 267, 268, 272, 276, 290, 291, 295, 317, 334, 335, 340, 343, 344

Offset: 1

Author

Hugo Pfoertner, Apr 04 2003

Keywords

• nonn

Comments

Let p and q be distinct odd primes, and take a and b so that a*p - b*q = 1. Dickson's conjecture implies there are infinitely many k such that 2*a+k*q and 2*b+k*p are prime, in which case n = a*p + (k*q*p-1)/2 is in the sequence with 2*n-1 = (2*b+k*p)*q and 2*n+1 = (2*a+k*q)*p. - Robert Israel, Aug 13 2018

Examples

			17 is a term because 2*17 - 1 = 33 = 3*11 and 2*17 + 1 = 35 = 5*7 are both semiprimes.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Wikipedia, Dickson's conjecture.

Crossrefs

Cf. A001358, A082131.

Programs

Maple

OSP:= select(numtheory:-bigomega=2, {seq(i,i=3..1000,2)}):
R:= map(t -> (t+1)/2, OSP intersect map(`-`,OSP,2)):
sort(convert(R,list)); # Robert Israel, Aug 13 2018

PARI

isok(n) = (bigomega(2*n-1) == 2) && (bigomega(2*n+1) == 2); \\ Michel Marcus, Jul 16 2017

Python

from sympy import factorint
from itertools import count, islice
def agen(): # generator of terms
    nxt = 0
    for k in count(2, 2):
        prv, nxt = nxt, sum(factorint(k+1).values())
        if prv == nxt == 2: yield k//2
print(list(islice(agen(), 56))) # Michael S. Branicky, Nov 26 2022

Extensions

More terms from Jud McCranie, Apr 04 2003