cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A110025 Smallest primes starting a complete three iterations Cunningham chain of the first or second kind.

Original entry on oeis.org

509, 1229, 1409, 2131, 2311, 2699, 3539, 6211, 6449, 7411, 10321, 10589, 11549, 11909, 12119, 17159, 18121, 19709, 19889, 22349, 22531, 23011, 24391, 26189, 27479, 29671, 30389, 31771, 35311, 41491, 43649, 46411, 54601, 55229, 56311
Offset: 1

Views

Author

Alexandre Wajnberg, Sep 03 2005

Keywords

Comments

Terms computed by Gilles Sadowski.

Examples

			1409 is here because, through the operator <2p+1> for chains of the first kind, 1409 -> 2819 -> 5639 -> 11279 and the chain ends here.
2131 is here because, through the operator <2p-1> for chains of the second kind, 2131 -> 4261 -> 8521 -> 17041 and the chain ends here.

Links

Crossrefs

Cf. A023272, A023302, A023330, A005384, A005385, A059452, A059455, A007700.
Cf. A059759, A059760, A059761, A059762, A059763, A059764, A059765, A038397, A104349, A091314, A069362, A016093, A014937, A057326.

Formula

Union of A059763 and A110024. [R. J. Mathar, May 08 2009]

Extensions

Edited by R. J. Mathar, May 08 2009

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

Original entry on oeis.org

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

Views

Author

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

Keywords

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.

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

Original entry on oeis.org

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

Views

Author

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

Keywords

Comments

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)

Examples

			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

Crossrefs

Cf. A005602, A059761, A059762, A110024 , A176379

Programs

  • Maple
    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

Views

Author

Alexandre Wajnberg, Sep 04 2005

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

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.

Formula

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

Extensions

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

Views

Author

Alexandre Wajnberg, Sep 04 2005

Keywords

Comments

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).

Examples

			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.

Links

Crossrefs

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.

Extensions

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

Views

Author

Alexandre Wajnberg, Sep 04 2005

Keywords

Comments

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).

Examples

			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).

Links

Crossrefs

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.

Extensions

Terms computed by Gilles Sadowski.
Showing 1-6 of 6 results.

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