A005849 Indices of prime Cullen numbers: numbers k such that k*2^k + 1 is prime.

1, 141, 4713, 5795, 6611, 18496, 32292, 32469, 59656, 90825, 262419, 361275, 481899, 1354828, 6328548, 6679881

Offset: 1

Author

N. J. A. Sloane

Keywords

• hard
• nonn
• nice
• more

Comments

From Amiram Eldar, Jun 05 2021: (Start)

The terms were found by:

a(1) - Cullen (1905). He found that there are no other terms up to 100 with the possible exception of 53. Cunningham (1906) showed that the 53rd Cullen number is composite and that the only possible term up to 200 is 141.

a(2) - Robinson (1958).

a(3)-a(6) - Keller (1995).

a(7)-a(8) - Masakatu Morii (1997).

a(9)-a(10) - Jeffrey Young (1997).

a(11)-a(12) - Darren Smith (1998).

a(13) - Masakatu Morii (1998).

a(14) - Mark Rodenkirch (2005).

a(15) - Dennis R. Gesker (2009).

a(16) - Magnus Bergman (2009). (End)

References

• A. J. Cunningham, Solution of question 15897, Math. Quest. Educ. Times, Vol. 10 (1906), pp. 44-47.
• Jean-Marie De Koninck, Ces nombres qui nous fascinent, Entry 141, p. 48, Ellipses, Paris 2008.
• Harvey Dubner, Generalized Cullen numbers, J. Rec. Math., Vol. 21, No. 3 (1989), pp. 190-191.
• R. K. Guy, Unsolved Problems in Number Theory, Springer, 1st edition, 1981. See section B20.
• Paulo Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 283.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Ray Ballinger, Cullen Primes: Definition and Status.
• Chris K. Caldwell, The Top Twenty: Cullen Primes.
• Allan Cunningham and H. J. Woodall, Factorisation of Q = (2^q -/+ q) and (q*2^q -\+ 1), The Messenger of Mathematics, Vol. 47 (1917-18), pp. 1-38. See p. 22.
• Harvey Dubner, Generalized Cullen numbers, J. Rec. Math., Vol. 21, No. 3 (1989), pp. 190-191. (Annotated scanned copy)
• Wilfrid Keller, New Cullen primes, Mathematics of Computation, Vol. 64, No. 212 (1995), pp. 1733-1741.
• Rudolf Ondrejka, The Top Ten: a Catalogue of Primal Configurations.
• PrimeGrid, PrimeGrid Primes: Subproject: (CUL) Cullen Prime Search.
• Raphael M. Robinson, A Report on primes of the form k*2^n + 1 and on factors of Fermat numbers, Proceedings of the American Mathematical Society, Vol. 9, No. 5 (1958), pp. 673-681.
• Eric Weisstein's World of Mathematics, Cullen Number.
• Eric Weisstein's World of Mathematics, Integer Sequence Primes.

Crossrefs

Cf. A002064, A002234, A050920, A173474 (complement).

Programs

Mathematica

Select[Range[1000], PrimeQ[# 2^# + 1] &] (* Alonso del Arte, Jul 30 2017 *)

PARI

is(n)=isprime(n<Charles R Greathouse IV, Feb 06 2017

Extensions

a(14) = 1354828 from old Proth Search pages by Mohammed Bouayoun (mohammed.bouayoun(AT)sanef.com), Apr 20 2006

The term 1467763 was added in error and has now been deleted; Jens Kruse Andersen, Nov 28 2007, remarks that 1467763 * 2^1467763 - 1 is a Woodall prime, but 3 divides the Cullen number 1467763 * 2^1467763 + 1.

6328548 from John Blazek, May 14 2009. He later reports that the search of the range from 6300000 to 6328548 was completed on May 28 2009.

Added a(16) = 6679881 from Caldwell's page, fixed broken link. - M. F. Hasler, Jan 18 2015

Name edited by Andrey Zabolotskiy and Felix Fröhlich, May 28 2021