A176379 -id:A176379 - cp's OEIS frontend

A176466 The smallest prime q which stays prime through at least 3 iterations of q -> 2 * q + prime(n+1).

2, 13, 5, 199, 2, 13, 251, 487, 61, 5, 113, 19, 2, 13, 157, 1621, 269, 23, 139, 557, 5, 37, 241, 5, 19, 587, 823, 41, 97, 5, 109, 13, 1151, 31, 1409, 53, 5, 1543, 67, 421, 5, 1039, 2, 13, 41, 359, 1697, 43, 101, 157, 1531, 179, 79, 193, 37, 181, 149, 113, 4519, 197, 397, 23, 739, 2, 283, 29, 5, 163, 1031, 1987

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 18 2010

See comments and references of A176379.
q, 2 * q + prime(n+1), 4 * q + 3 * prime(n+1) and 8 * q + 7 * prime(n+1) are required to be prime.
List of (q,first iteration, 2nd iteration, 3rd iteration):
(2,7,17,37) (13,31,67,139) (5,17,41,89) (199,409,829,1669) (2,17,47,107)
(13,43,103,223) (251,521,1061,2141) (487,997,2017,4057) (61,151,331,691) (5,41,113,257)
(113,263,563,1163) (19,79,199,439) (2,47,137,317) (13,73,193,433) (157,367,787,1627)
(1621,3301,6661,13381) (269,599,1259,2579) (23,113,293,653) (139,349,769,1609) (557,1187,2447,4967)
(5,89,257,593) (37,157,397,877) (241,571,1231,2551) (5,107,311,719) (19,139,379,859)
(587,1277,2657,5417) (823,1753,3613,7333) (41,191,491,1091) (97,307,727,1567) (5,137,401,929)
(109,349,829,1789) (13,163,463,1063) (1151,2441,5021,10181) (31,211,571,1291) (1409,2969,6089,12329)
(53,263,683,1523) (5,173,509,1181) (1543,3253,6673,13513) (67,307,787,1747) (421,1021,2221,4621)
(5,191,563,1307) (1039,2269,4729,9649) (2,197,587,1367) (13,223,643,1483) (41,281,761,1721)
(359,929,2069,4349) (1697,3617,7457,15137) (43,313,853,1933) (101,431,1091,2411) (157,547,1327,2887)

			n=1: q=2, iteration 2 * q + prime(2) = 7, iteration 2 * 7 + 3 = 17, 2 * 17 + 3 = 37: q=2 is first term
n=2: q=13, iteration 2 * 13 + prime(3) = 31, iteration 2 * 31 + 5 = 67, iteration 2 * 67 + 5 = 139, q=13 is 2nd term

Cf. A005602, A059761, A059762, A110024 , A176379

A176466 := 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 ;
        end if;
        pitr := 2*pitr+pk1 ;
        if not isprime(pitr) then
            next ;
        else
            return p ;
        end if;
    end do:
end proc:
seq(A176466(n),n=1..80) ; # R. J. Mathar, May 21 2025

cp's OEIS Frontend

A176466 The smallest prime q which stays prime through at least 3 iterations of q -> 2 * q + prime(n+1).

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