A050920 -id:A050920 - cp's OEIS frontend

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

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)

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).

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.

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

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

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

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

A210339 Generalized Cullen primes: any primes that can be written in the form n*b^n + 1 with n+2 > b > 2.

Original entry on oeis.org

19, 193, 52489, 114689, 9000000001, 259374246011, 38280596832649217, 59296646043258913, 408700964355468751, 2434970217729660813313, 13576803638250229989377, 21000000000000000000001, 3140085798164163223281069127, 4818833289797717549937328129

Offset: 1

Arkadiusz Wesolowski, Mar 20 2012

nonn

			81*2^324 + 1 is a prime number and 81*2^324 + 1 = 81*16^81 + 1, so this number is in the sequence.

Harvey Dubner, Generalized Cullen numbers, J. Recreational Math. 21 (1989), pp. 190-194.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..92
Chris Caldwell, The Top 20 Generalized Cullen Primes
Chris Caldwell, The Prime Glossary, Cullen prime
G. L. Honaker, Jr. and Chris Caldwell, Prime Curios! 57
G. L. Honaker, Jr. and Chris Caldwell, 15545...00001 (1008-digits)
G. L. Honaker, Jr. and Chris Caldwell, 36869...81251 (10002-digits)
PrimeGrid, Home Page
Wikipedia, Cullen number

Cf. A050920, A005849, A006552, A007646.

lst = {}; Do[p = n*b^n + 1; If[p < 10^200 && PrimeQ[p], AppendTo[lst, p]], {b, 3, 100}, {n, b - 1, 413}]; Sort@lst

A173474 Numbers n such that n*2^n + 1 is not prime.

Original entry on oeis.org

0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72

Offset: 1

Juri-Stepan Gerasimov, Feb 19 2010

nonn

Complement of "prime Cullen numbers" A005849.
Where a(n)=n for n <= 140, and a(141)=142,..., a(4711)=4712, a(4712)=4714,..., a(5792)=5794, a(5793)=5796,..., a(6607)=6610, a(6608)=6612,..., a(18491)=18495, a(18492)=18497,..., a(32286)=32291, a(32287)=32293,..., a(32462)=32468, a(32463)=32470,..., a(59648)=59655, a(59649)=59657,..., a(90816)=90824, a(90817)=90826,..., a(262403)=262418, a(262404)=262420,..., a(361264)=361274, a(361265)=361276,..., a(481887)=481898, a(481888)=481900,..., a(1354815)=1354827, a(1354816)=1354829,..., a(6328534)=6328547, a(6328535)=6328549,...
Otherwise said, this includes all nonnegative integers except for the "prime Cullen numbers" (more precisely, indices of primes in A002064): 1, 140, 4713, 5795, ... listed in A005849. - M. F. Hasler, Jan 18 2015

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A005849, A045344, A050920, A087156, A141468.

nnnpQ[n_]:=Module[{c=n 2^n+1},!PrimeQ[c]&&c>=0]; Select[Range[0,100], nnnpQ] (* Harvey P. Dale, Aug 23 2011 *)

Corrected and edited by M. F. Hasler, Jan 18 2015

Name edited by Michel Marcus, Nov 02 2017

A191568 Numbers k such that k*(k+1)^k+1 is prime.

Original entry on oeis.org

1, 2, 3, 9, 10, 14, 33, 36

Offset: 1

Juri-Stepan Gerasimov, Jun 13 2011

The corresponding primes are a subset of the generalized Cullen primes (A210339 U A050920). - Jeppe Stig Nielsen, Apr 12 2016

			a(1)=1 because 1*2^1+1=3 is prime, a(2)=2 because 2*3^2+1=19 is prime, a(3)=3 because 3*4^3+1=193 is prime, a(4)=9 because 9*10^9+1=9000000001 is prime, a(5)=10 because 10*11^10+1=259374246011 is prime.

Cf. A109618, A110709, A210339, A271718.

Select[Range@ 100, PrimeQ[# (# + 1)^# + 1] &] (* Michael De Vlieger, Apr 12 2016 *)

is(n)=ispseudoprime(n*(n+1)^n+1) \\ Charles R Greathouse IV, Jun 13 2017

A137716 Number of digits in the decimal expansion of the n-th Cullen prime.

Original entry on oeis.org

1, 45, 1423, 1749, 1994, 5573, 9726, 9779, 17964, 27347, 79002, 108761, 145072, 407850, 1905090, 2010852

Offset: 1

Ant King, Feb 09 2008

Cullen primes are prime numbers of the form k*2^k+1. This sequence is complete for all values of n up to 3500000.

			As the sixth Cullen prime, 18496*2^18496 + 1 = 1.311...*10^5572, is a 5573-digit number, we have a(6) = 5573.

Ray Ballinger and Mark Rodenkirch, Cullen Primes: Definition and Status.
James Cullen, Question 15897, The Educational Times, Dec. 1905, p. 534.
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.
Wilfrid Keller, New Cullen Primes, Mathematics of Computation, Vol. 64, No. 212 (Ocober 1995), pp. 1733-1741.

Cf. A055642, A005849, A002064.

a(n) = A055642(A050920(n)). [Corrected by Georg Fischer, Nov 18 2023]

a(15)-a(16) from Amiram Eldar, Oct 27 2024

A288159 a(n) = smallest k such that kn2^n+1 is prime.

Original entry on oeis.org

1, 2, 3, 3, 4, 2, 3, 6, 4, 4, 9, 3, 3, 5, 9, 7, 3, 3, 3, 5, 3, 7, 19, 5, 5, 2, 3, 7, 7, 9, 5, 15, 3, 10, 10, 7, 14, 6, 8, 6, 25, 6, 50, 45, 13, 4, 18, 31, 27, 2, 4, 33, 18, 2, 41, 18, 10, 9, 6, 7, 3, 32, 4, 39, 18, 17, 11, 30, 17, 18, 7, 51, 38, 11, 15, 13, 9, 10, 24, 2

Offset: 1

Pierre CAMI, Jun 06 2017

nonn

If k = 1 then n*2^n+1 is a Cullen prime (A050920).

Pierre CAMI, Table of n, a(n) for n = 1..10000

Cf. A050920, A288158.

Table[k = 1; While[! PrimeQ[k n*2^n + 1], k++]; k, {n, 80}] (* Michael De Vlieger, Jun 09 2017 *)

a(n) = my(k=1); while (! isprime(k*n*2^n+1), k++); k; \\ Michel Marcus, Jun 07 2017

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

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A210339 Generalized Cullen primes: any primes that can be written in the form n*b^n + 1 with n+2 > b > 2.

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A173474 Numbers n such that n*2^n + 1 is not prime.

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A191568 Numbers k such that k*(k+1)^k+1 is prime.

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A137716 Number of digits in the decimal expansion of the n-th Cullen prime.

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A288159 a(n) = smallest k such that k*n*2^n+1 is prime.

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A288159 a(n) = smallest k such that kn2^n+1 is prime.