A082130 - OEIS

%I #22 Nov 26 2022 12:35:07

%S 17,25,28,43,46,47,60,61,71,72,80,92,93,101,102,107,108,109,110,118,

%T 124,133,144,145,150,151,152,160,161,164,170,196,197,206,207,208,223,

%U 226,235,236,258,259,264,267,268,272,276,290,291,295,317,334,335,340,343,344

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

%C 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

%H Robert Israel, <a href="/A082130/b082130.txt">Table of n, a(n) for n = 1..10000</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Dickson%27s_conjecture">Dickson's conjecture</a>.

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

%p OSP:= select(numtheory:-bigomega=2, {seq(i,i=3..1000,2)}):

%p R:= map(t -> (t+1)/2, OSP intersect map(`-`,OSP,2)):

%p sort(convert(R,list)); # _Robert Israel_, Aug 13 2018

%o (PARI) isok(n) = (bigomega(2*n-1) == 2) && (bigomega(2*n+1) == 2); \\ _Michel Marcus_, Jul 16 2017

%o (Python)

%o from sympy import factorint

%o from itertools import count, islice

%o def agen(): # generator of terms

%o     nxt = 0

%o     for k in count(2, 2):

%o         prv, nxt = nxt, sum(factorint(k+1).values())

%o         if prv == nxt == 2: yield k//2

%o print(list(islice(agen(), 56))) # _Michael S. Branicky_, Nov 26 2022

%Y Cf. A001358, A082131.

%K nonn

%O 1,1

%A _Hugo Pfoertner_, Apr 04 2003

%E More terms from _Jud McCranie_, Apr 04 2003