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
KEYWORD
nonn
AUTHOR
Hugo Pfoertner, Apr 04 2003
EXTENSIONS
More terms from Jud McCranie, Apr 04 2003
STATUS
approved
