A059761 -id:A059761 - 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

A110089 Smallest prime beginning (through <2+1> or/and <2-1>) a complete Cunningham chain (of the first or the second kind) of length n.

Original entry on oeis.org

11, 3, 2, 509, 2, 89, 16651, 15514861, 85864769, 26089808579, 665043081119, 554688278429, 758083947856951, 95405042230542329, 69257563144280941

Offset: 1

Alexandre Wajnberg, Sep 04 2005

The word "complete" indicates each chain is exactly n primes long for the operator in function (i.e. the chain cannot be a subchain of another one); and the first and/or last term may be involved in a chain of the other kind (i.e. the chain may be connected to another one). a(1)-a(8) computed by Gilles Sadowski.

			a(1)=11 because 2, 3, 5 and 7 are included in longer chains than one prime long; and 11 (although included in a <2p+1> chain) has no prime connection through <2p-1>.
a(2)=3 because 3 begins (through 2p+1>) the first complete two primes chain: 3-> 7 (even if 3 and 7 are also part of two others chains, but through <2p-1>).
a(3)=2 because (although 2 begins also a five primes chain through <2p+1>) it begins, through <2p-1>, the first complete three primes chain encountered: 2->3->5.

Chris Caldwell's Prime Glossary, Cunningham chains.

Cf. A023272, A023302, A023330, A005384, A005385, A059452, A059455, A007700, A059759, A059760, A059761, A059762, A059763, A059764, A059765, A038397, A104349, A091314, A069362, A016093, A014937, A057326, A110059, A110056, A110038, A059766, A110027, A059764, A110025, A110024, A059763, A110022, A109998, A109946, A109927, A109835, A005603.

a(n) = min(A005602(n), A005603(n)). - R. J. Mathar, Jul 23 2008

a(8)-a(13) via A005602, A005603 from R. J. Mathar, Jul 23 2008

a(14)-a(15) via A005602, A005603 from Jason Yuen, Sep 03 2024

A110092 Smallest prime ending (through <2+1> or <2-1> separately) a complete Cunningham chain (of the first or the second kind) of length n.

Original entry on oeis.org

17, 59, 73, 4079, 47, 2879, 1065601

Offset: 1

Alexandre Wajnberg, Sep 04 2005

The word "complete" indicates each chain is exactly n primes long for the operator in function (i.e. the chain cannot be a subchain of another one); and the first and/or last term may not be involved in a chain of the other kind (i.e. the chain may not be connected to another one).

			a(1)=17 because 2, 3, 5, 7, 11 and 13 are part of longer chains whatever the operator; 17 is the first completely isolated prime.
a(2)=59 because it ends the first two primes chain not connected to another one: 29->59.

Chris Caldwell's Prime Glossary, Cunningham chains.

Cf. A023272, A023302, A023330, A005384, A005385, A059452, A059455, A007700, Cf. A059759, A059760, A059761, A059762, A059763, A059764, A059765, A038397, A104349, A091314, A069362, A016093, A014937, A057326, A110059, A110056, A110038, A059766, A110027, A059764, A110025, A110024, A059763, A110022, A109998, A109946, A109927, A109835, A005603.

Terms computed by Gilles Sadowski.

A110093 Smallest prime ending (through <2+1> or/and <2-1>) a complete Cunningham chain (of the first or the second kind) of length n.

Original entry on oeis.org

11, 7, 5, 4079, 47, 2879, 1065601

Offset: 1

Alexandre Wajnberg, Sep 04 2005

The word "complete" indicates each chain is exactly n primes long for the operator in function (i.e. the chain cannot be a subchain of another one); but the first and/or last term may be involved in a chain of the other kind (i.e. the chain may be connected to another one).

			a(1)=11 because 2, 3, 5 and 7 are not ending chains; or are part of chains longer than one prime; 11, although is part of a five primes <2p+1> chain, is isolated through <2p-1>.
a(2)=7 because 7 ends through <2p+1> the first two primes chain: 3->7 (even if both primes are also part of <2p-1> chains).

Chris Caldwell's Prime Glossary, Cunningham chains.

Cf. A023272, A023302, A023330, A005384, A005385, A059452, A059455, A007700, Cf. A059759, A059760, A059761, A059762, A059763, A059764, A059765, A038397, A104349, A091314, A069362, A016093, A014937, A057326, A110059, A110056, A110038, A059766, A110027, A059764, A110025, A110024, A059763, A110022, A109998, A109946, A109927, A109835, A005603.

Terms computed by Gilles Sadowski.

A320393 First members of the Cunningham chains of the first kind whose length is a prime.

Original entry on oeis.org

2, 3, 11, 23, 29, 41, 53, 83, 113, 131, 173, 179, 191, 233, 239, 251, 281, 293, 419, 431, 443, 491, 593, 641, 653, 659, 683, 719, 743, 761, 809, 911, 953, 1013, 1019, 1031, 1049, 1103, 1223, 1289, 1439, 1451, 1481, 1499, 1511, 1559, 1583, 1601, 1733, 1811, 1889, 1901, 1931, 1973, 2003, 2039, 2063, 2069, 2129, 2141

Offset: 1

Pierandrea Formusa, Dec 10 2018

nonn

			41 is an item as it generates the Cunningham chain (41, 83, 167), of length 3, that is prime.

Cf. A059761, A059762, A059764.

aQ[n_] := PrimeQ[Length[NestWhileList[2#+1&, n, PrimeQ]] - 1]; Select[Range[2200], aQ] (* Amiram Eldar, Dec 11 2018 *)

from sympy.ntheory import isprime
def cunningham_chain(p,t):
    #it returns the cunningham chain generated by p of type t (1 or 2)
    if not(isprime(p)):
        raise Exception("Invalid starting number! It must be prime")
    if t!=1 and t!=2:
        raise Exception("Invalid type! It must be 1 or 2")
    elif t==1: k=t
    else: k=-1
    cunn_ch=[]
    cunn_ch.append(p)
    while isprime(2*p+k):
        p=2*p+k
        cunn_ch.append(p)
    return(cunn_ch)
from sympy import prime
n=350
r=""
for i in range(1,n):
    cunn_ch=(cunningham_chain(prime(i),1))
    lcunn_ch=len(cunn_ch)
    if isprime(lcunn_ch):
       r += ","+str(prime(i))
print(r[1:])

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

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A110089 Smallest prime beginning (through <*2+1> or/and <*2-1>) a complete Cunningham chain (of the first or the second kind) of length n.

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A110092 Smallest prime ending (through <*2+1> or <*2-1> separately) a complete Cunningham chain (of the first or the second kind) of length n.

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A110093 Smallest prime ending (through <*2+1> or/and <*2-1>) a complete Cunningham chain (of the first or the second kind) of length n.

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A320393 First members of the Cunningham chains of the first kind whose length is a prime.

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Python

A110089 Smallest prime beginning (through <2+1> or/and <2-1>) a complete Cunningham chain (of the first or the second kind) of length n.

A110092 Smallest prime ending (through <2+1> or <2-1> separately) a complete Cunningham chain (of the first or the second kind) of length n.

A110093 Smallest prime ending (through <2+1> or/and <2-1>) a complete Cunningham chain (of the first or the second kind) of length n.