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

1,1

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

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Wikipedia, Dickson's conjecture.

Example

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

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

Prog

(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

Crossrefs

Cf. A001358, A082131.

Sequence in context: A272635 A105448 A336007 * A140609 A131275 A227238

Adjacent sequences: A082127 A082128 A082129 * A082131 A082132 A082133

Keyword

nonn

Author

Hugo Pfoertner, Apr 04 2003

Extensions

More terms from Jud McCranie, Apr 04 2003

Status

approved