A005849 -id:A005849 - cp's OEIS frontend

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A367004 a(n) is the smallest prime factor of n*2^n+1.

3, 3, 5, 5, 7, 5, 3, 3, 11, 7, 13, 13, 3, 3, 17, 17, 5, 11, 3, 3, 23, 13, 5, 5, 3, 3, 7, 29, 31, 17, 3, 3, 47, 19, 37, 37, 3, 3, 41, 41, 13, 23, 3, 3, 11, 5, 7, 7, 3, 3, 53, 7, 5591, 29, 3, 3, 5, 31, 37, 61, 3, 3, 5, 5, 67, 5, 3, 3, 7, 37, 11, 41, 3, 3, 149

Offset: 1

Sean A. Irvine, Oct 31 2023

nonn

Amiram Eldar, Table of n, a(n) for n = 1..914 (terms 1..350 from Robert Israel)

Cf. A002064, A005849, A020639, A049479, A367002, A367003, A367005.

seq(min(numtheory:-factorset(n*2^n+1)), n=1..100); # Robert Israel, Nov 09 2023

Table[FactorInteger[n 2^n+1][[1,1]],{n,80}] (* Harvey P. Dale, Aug 14 2024 *)

a(n) = A020639(A002064(n)).

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

A357612 Numbers k such that 1 + 2^k*k^3 is prime.

Original entry on oeis.org

1, 5, 41, 202, 281, 394, 1157, 1211, 1816, 9845, 19780, 50800, 98621, 101945

Offset: 1

Juri-Stepan Gerasimov, Nov 17 2022

			1 is in this sequence because 1 + 2^1*1^3 = 5 is prime;
5 is in this sequence because 1 + 2^5*5^3 = 4001 is prime.

Cf. A000040, A005849, A058780.

[n: n in [1..2000] | IsPrime(1+2^n*n^3)];

Select[Range[2000], PrimeQ[1 + 2^# * #^3] &] (* Amiram Eldar, Nov 17 2022 *)

a(11)-a(12) from Amiram Eldar, Nov 17 2022

a(13)-a(14) from Michael S. Branicky, May 16 2023

A007637 Primes of form 3k^2 - 3k + 23.

Original entry on oeis.org

23, 29, 41, 59, 83, 113, 149, 191, 239, 293, 353, 419, 491, 569, 653, 743, 839, 941, 1049, 1163, 1283, 1409, 1823, 1973, 2129, 2459, 2633, 2999, 3191, 3389, 3593, 3803, 4019, 4241, 4703, 4943, 5189, 5441, 6791, 7079, 7673, 8291, 8609

Offset: 1

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

Paul Hoffman, Archimedes' Revenge, Penguin Books 1988, pp. 39-40.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 145.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Harvey Dubner, Generalized Cullen numbers, J. Rec. Math., 21 (No. 3, 1989), 190-191. (Annotated scanned copy)

Cf. A007638.

[a: n in [1..60] | IsPrime(a) where a is 3*n^2-3*n+23]; // Vincenzo Librandi, Mar 20 2013

Select[Table[3 n^2 - 3 n + 23, {n, 60}], PrimeQ] (* Vincenzo Librandi, Mar 20 2013 *)

A007639 Primes of form 2n^2 - 2n + 19.

Original entry on oeis.org

19, 23, 31, 43, 59, 79, 103, 131, 163, 199, 239, 283, 331, 383, 439, 499, 563, 631, 859, 1031, 1123, 1319, 1423, 1531, 1759, 1879, 2003, 2131, 2399, 2539, 2683, 3299, 3463, 3631, 3803, 4159, 4723, 4919, 5119, 5323, 5531, 5743, 6863, 7583, 8599

Offset: 1

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
H. Dubner, Generalized Cullen numbers, J. Rec. Math., 21 (No. 3, 1989), 190-191. (Annotated scanned copy)

Cf. A139827.

[a: n in [1..60] | IsPrime(a) where a is 2*n^2-2*n+19]; // Vincenzo Librandi, Mar 20 2013

Select[Table[2n^2-2n+19,{n,90}],PrimeQ] (* Harvey P. Dale, Dec 19 2011 *)

The primes are congruent to {2, 15, 19, 23, 31, 35, 39, 43, 51, 55, 59, 79, 87, 91, 103, 119, 131, 135, 143} (mod 148). - T. D. Noe, May 02 2008

A242175 Numbers k such that k*2^k + 1 is a semiprime.

Original entry on oeis.org

2, 3, 4, 5, 8, 9, 11, 16, 21, 33, 35, 101, 105, 131, 158, 165, 191, 234, 251, 435, 453, 459, 561, 579, 604, 671, 744, 753, 933, 963, 1041, 1146, 1168, 1254, 1794

Offset: 1

Vincenzo Librandi, May 07 2014

The semiprimes of this form are 9, 25, 65, 161, 2049, 4609, 22529, ... (A242116).
a(35) >= 1528. Below 2000, 1794 and 1961 are in the sequence. Unknown factorization for 1528, 1576, 1908. - Hugo Pfoertner, Jul 29 2019
The k*2^k + 1 corresponding to 1528 and 1576 each have at least three prime factors. - Tyler Busby, Mar 16 2025

FactorDB, Status of 1908*2^1908+1.

Cf. A001358, A002064, A005849, A242116, A242273.

IsSemiprime:=func; [n: n in [2..230] | IsSemiprime(s) where s is n*2^n+1]; // Bruno Berselli, May 08 2014

Mathematica

Select[Range[165], Plus@@Last/@FactorInteger[# * 2^# + 1]==2&]

Formula

A002064(a(n)) = A242116(n). - Amiram Eldar, Nov 27 2019

Extensions

a(17) from Bruno Berselli, May 08 2014

a(18)-a(30) from Luke March, Aug 13 2015

a(31)-a(34) from Hugo Pfoertner, Jul 29 2019

Wrong term 941 removed by Amiram Eldar, Nov 27 2019

a(35) from Tyler Busby, Mar 16 2025

A366422 Numbers k such that k^4*2^k + 1 is a prime.

Original entry on oeis.org

1, 24, 33, 36, 99, 195, 244, 464, 567, 621, 741, 1395, 2164, 3309, 3537, 3708, 4413, 5001, 5187, 5292, 15504, 18816, 19521, 24657, 27972, 57687

Offset: 1

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Author

Juri-Stepan Gerasimov, Nov 16 2023

Keywords

nonn
more

Comments

No further terms <= 100000. - Michael S. Branicky, Nov 17 2023

Crossrefs

Numbers k such that k^m*2^k + 1 is a prime: 0, 1, 2, 4, 8, 16, .. (m = 0), A005849 (m = 1), A058780 (m = 2), A357612 (m = 3), this sequence (m = 4).

Cf. A092506, A367102.

Programs

Magma

[k: k in [0..4000] | IsPrime(k^4*2^k+1)];

Mathematica

Select[Range[6000], PrimeQ[#^4*2^# + 1] &] (* Amiram Eldar, Nov 16 2023 *)

Extensions

a(22)-a(25) from Amiram Eldar, Nov 17 2023

a(26) from Michael S. Branicky, Nov 17 2023

A367421 Numbers k such that k^5*2^k + 1 is a prime.

Original entry on oeis.org

1, 41, 53, 231, 532, 1632, 1642, 9701, 13372, 19613, 25518, 31929, 92476, 97433

Offset: 1

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Author

Juri-Stepan Gerasimov, Nov 18 2023

Keywords

nonn
more

Crossrefs

Numbers k such that k^m*2^k + 1 is a prime: 0, 1, 2, 4, 8, 16, .. (m = 0), A005849 (m = 1), A058780 (m = 2), A357612 (m = 3), A366422 (m = 4), this sequence (m = 5).

Programs

Magma

[k: k in [1..1000] | IsPrime(k^5*2^k+1)];

Mathematica

Select[Range[2000], PrimeQ[#^5*2^# + 1] &] (* Amiram Eldar, Nov 18 2023 *)

Extensions

a(10)-a(12) from Michael S. Branicky, Nov 18 2023

a(13)-a(14) from Michael S. Branicky, Aug 26 2024

A007638 Numbers k such that 3*k^2 - 3*k + 23 is composite.

Original entry on oeis.org

23, 24, 28, 31, 39, 44, 45, 46, 47, 50, 52, 56, 57, 60, 63, 67, 69, 70, 71, 79, 80, 85, 86, 88, 89, 90, 92, 93, 96, 97, 102, 107, 108, 112, 115, 116, 118, 119, 121, 122, 123, 126, 128, 131, 134, 137, 138, 139, 143, 144, 145, 147, 148, 151, 153, 156, 157, 161, 162

Offset: 1

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Author

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

Keywords

nonn
easy

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..3100

H. Dubner, Generalized Cullen numbers, J. Rec. Math., 21 (No. 3, 1989), 190-191. (Annotated scanned copy)

Programs

Magma

[n: n in [0..200]| not IsPrime(3*n^2-3*n+23)]; // Vincenzo Librandi, Jun 08 2017

Mathematica

Select[Range[200], !PrimeQ[3 #^2 - 3 # + 23] &] (* Vincenzo Librandi, Jun 08 2017 *)

PARI

is(n)=!isprime(3*n^2 - 3*n + 23) \\ Charles R Greathouse IV, Oct 23 2015

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

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Author

Juri-Stepan Gerasimov, Feb 19 2010

Keywords

nonn

Comments

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

Links

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

Crossrefs

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

Programs

Mathematica

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

Extensions

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

Name edited by Michel Marcus, Nov 02 2017

Previous Showing 11-20 of 29 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.
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cp's OEIS Frontend

A367004 a(n) is the smallest prime factor of n*2^n+1.

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Maple

Mathematica

Formula

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

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

A357612 Numbers k such that 1 + 2^k*k^3 is prime.

Views

Author

Keywords

Examples

Crossrefs

Programs

Magma

Mathematica

Extensions

A007637 Primes of form 3*k^2 - 3*k + 23.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Magma

Mathematica

A007639 Primes of form 2n^2 - 2n + 19.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A242175 Numbers k such that k*2^k + 1 is a semiprime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Formula

Extensions

A366422 Numbers k such that k^4*2^k + 1 is a prime.

Views

Author

Keywords

Comments

Crossrefs

Programs

Magma

Mathematica

Extensions

A367421 Numbers k such that k^5*2^k + 1 is a prime.

Views

Author

Keywords

Crossrefs

Programs

A007637 Primes of form 3k^2 - 3k + 23.

A007638 Numbers k such that 3k^2 - 3k + 23 is composite.