A176379 The smallest prime q which stays prime through at least two iterations of q -> := 2 * q + prime(n+1).

2, 7, 2, 31, 2, 7, 11, 7, 19, 5, 5, 19, 2, 13, 13, 61, 11, 17, 61, 5, 5, 7, 139, 5, 19, 2, 103, 29, 7, 2, 109, 7, 59, 31, 41, 5, 5, 127, 13, 31, 5, 109, 2, 7, 41, 11, 2, 7, 101, 67, 79, 5, 31, 13, 37, 19, 11, 2, 109, 53, 7, 2, 19, 2, 127, 29, 5, 13, 59, 7, 19, 47, 47, 11, 13, 79, 17, 19, 89, 619

Offset: 1

Author

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 16 2010

Keywords

• base
• nonn

Comments

Such q are generalized Cunningham primes: prime numbers p(1), ..., p(n):

q, f(q) = 2 * q + prime(n+1), f(f(q)) = 4 * q + 3 * prime(n+1) must be primes.

n = 0 is omitted as the first iteration 2 * q + prime(1) = 2 * (q+1) is generally even.

List of (q, first iteration, second iteration):

(2,7,17) (7,19,43) (2,11,29) (31,73,157) (2,17,47)

(7,31,79) (11,41,101) (7,37,97) (19,67,163) (5,41,113)

(5,47,131) (19,79,199) (2,47,137) (13,73,193) (13,79,211)

(61,181,421) (11,83,227) (17,101,269) (61,193,457) (5,83,239)

(5,89,257) (7,97,277) (139,367,823) (5,107,311) (19,139,379)

(2,107,317) (103,313,733) (29,167,443) (7,127,367) (2,131,389)

(109,349,829) (7,151,439) (59,257,653) (31,211,571) (41,233,617)

(5,167,491) (5,173,509) (127,421,1009) (13,199,571) (31,241,661)

(5,191,563) (109,409,1009) (2,197,587) (7,211,619) (41,281,761)

(11,233,677) (2,227,677) (7,241,709) (101,431,1091) (67,367,967)

Examples

			n=1, prime(n+1) = 3: checking q=2: 2 * 2 + 3 = 7, 2 * 7 + 3 = 17, q=2 is first term.
n=2: checking q=7: 2 * 7 + 5 = 19, 2 * 19 + 5 = 43, 7 is 2nd term.
n=3: checking q=2: 2 * 2 + 7 = 11, 2 * 11 + 7 = 29, 2 is 3rd term.
		

References

• Joe Buhler, Algorithmic Number Theory, Third International Symposium, ANTS-III, Springer New York, 1998.
• R. K. Guy, Unsolved problems in number theory, Springer-Verlag, New York, 1994.
• Paulo Ribenboim, Die Welt der Primzahlen, Geheimnisse und Rekorde, Springer-Verlag GmbH & Co. KG, 2006.

Crossrefs

Cf. A005602, A059761, A059762, A110024.

Programs

Maple

A176379 := proc(n)
    pk1 := ithprime(n+1) ;
    for pidx from 1 do
        p := ithprime(pidx) ;
        pitr := 2*p+pk1 ;
        if not isprime(pitr) then
            next ;
        end if;
        pitr := 2*pitr+pk1 ;
        if not isprime(pitr) then
            next ;
        else
            return p ;
        end if;
    end do:
end proc:
seq(A176379(n),n=1..20) ; # R. J. Mathar, May 21 2025

Formula

a(n) = smallest prime q such that 2*q+prime(n+1) is prime and 2*(2*q+prime(n+1))+prime(n+1) is also prime.