A002254 -id:A002254 - cp's OEIS frontend

A002253 Numbers k such that 3*2^k + 1 is prime.

1, 2, 5, 6, 8, 12, 18, 30, 36, 41, 66, 189, 201, 209, 276, 353, 408, 438, 534, 2208, 2816, 3168, 3189, 3912, 20909, 34350, 42294, 42665, 44685, 48150, 54792, 55182, 59973, 80190, 157169, 213321, 303093, 362765, 382449, 709968, 801978, 916773, 1832496, 2145353

Offset: 1

N. J. A. Sloane

From Zak Seidov, Mar 08 2009: (Start)
List is complete up to 3941000 according to the list of RB & WK.
So far there are only 4 primes: 2, 5, 41, 353. (End)

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 2, p. 614.
H. Riesel, "Prime numbers and computer methods for factorization", Progress in Mathematics, Vol. 57, Birkhauser, Boston, 1985, Chap. 4, see pp. 381-384.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Jeppe Stig Nielsen, Table of n, a(n) for n = 1..50
Ray Ballinger, Proth Search Page
Ray Ballinger and Wilfrid Keller, List of primes k.2^n + 1 for k < 300
C. K. Caldwell, The Prime Pages
Y. Gallot, Proth.exe: Windows Program for Finding Large Primes
S. W. Golomb, Properties of the sequence 3.2^n+1, Math. Comp., 30 (1976), 657-663.
S. W. Golomb, Properties of the sequence 3.2^n+1, Math. Comp., 30 (1976), 657-663. [Annotated scanned copy]
Wilfrid Keller, List of primes k.2^n - 1 for k < 300
PrimeGrid, 321 Prime Search statistics
R. M. Robinson, A report on primes of the form k.2^n+1 and on factors of Fermat numbers, Proc. Amer. Math. Soc., 9 (1958), 673-681. [Annotated scanned copy of selected pages. The first page is (accidentally) included with the scan of the Wilson letter below.]
R. G. Wilson v, Letter (FAX) to N. J. A. Sloane, May 20 1994, concerning A002253-A002274, A000043, A002235-A002240, A001770-A001775, A007117, and many other sequences
Eric Weisstein's World of Mathematics, Proth Prime
R. G. Wilson, V, Letter to N. J. A. Sloane, Jan. 1989
Index entries for sequences of n such that k*2^n-1 (or k*2^n+1) is prime

Cf. A002254, A004119, A181565.

See A039687 for the actual primes.

is(n)=isprime(3*2^n+1) \\ Charles R Greathouse IV, Feb 17 2017

A2253=[1]; A002253(n)=for(k=#A2253, n-1, my(m=A2253[k]); until(ispseudoprime(3<M. F. Hasler, Mar 03 2023

a(n) = log_2((A039687(n)-1)/3) = floor(log_2(A039687(n)/3)). - M. F. Hasler, Mar 03 2023

Corrected and extended according to the list of Ray Ballinger and Wilfrid Keller by Zak Seidov, Mar 08 2009
Edited by N. J. A. Sloane, Mar 13 2009
a(47) and a(48) from the Ballinger & Keller web page, Joerg Arndt, Apr 07 2013
a(49) from https://t5k.org/primes/page.php?id=116922, Fabrice Le Foulher, Mar 09 2014
Terms moved from Data to b-file (Links), and additional term appended to b-file, by Jeppe Stig Nielsen, Oct 30 2020

A226366 Numbers k such that 5*2^k + 1 is a prime factor of a Fermat number 2^(2^m) + 1 for some m.

Original entry on oeis.org

7, 25, 39, 75, 127, 1947, 3313, 23473, 125413

Offset: 1

Arkadiusz Wesolowski, Jun 05 2013

No other terms below 5330000.

The reason all terms are odd is that if k is even, then 5*2^k + 1 == (-1)*(-1)^k + 1 = (-1)*1 + 1 = 0 (mod 3). So if k is even, then 3 divides 5*2^k + 1, and since 3 divides no other Fermat number than F_0=3 itself, we do not have a Fermat factor. - Jeppe Stig Nielsen, Jul 21 2019

Wilfrid Keller, Fermat factoring status
J. C. Morehead, Note on the factors of Fermat's numbers, Bull. Amer. Math. Soc., Volume 12, Number 9 (1906), pp. 449-451.
Eric Weisstein's World of Mathematics, Fermat Number

Subsequence of A002254.

Cf. A000215, A050526, A057775, A057778, A201364, A204620.

lst = {}; Do[p = 5*2^n + 1; If[PrimeQ[p] && IntegerQ@Log[2, MultiplicativeOrder[2, p]], AppendTo[lst, n]], {n, 7, 3313, 2}]; lst

isok(n) = my(p = 5*2^n + 1, z = znorder(Mod(2, p))); isprime(p) && ((z >> valuation(z, 2)) == 1); \\ Michel Marcus, Nov 10 2018

A001770 Numbers k such that 5*2^k - 1 is prime.

Original entry on oeis.org

2, 4, 8, 10, 12, 14, 18, 32, 48, 54, 72, 148, 184, 248, 270, 274, 420, 1340, 1438, 1522, 1638, 1754, 1884, 2014, 2170, 2548, 2622, 2652, 2704, 13510, 21738, 25624, 41934, 51478, 52540, 53230, 172300, 245728, 350028, 1194164

Offset: 1

N. J. A. Sloane

A084213(a(n)+1) is in A136539, for all n. - Farideh Firoozbakht and M. F. Hasler, Nov 03 2012

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Y. Gallot, Proth.exe: Windows Program for Finding Large Primes
Wilfrid Keller, List of primes k.2^n - 1 for k < 300
H. C. Williams and C. R. Zarnke, A report on prime numbers of the forms M = (6a+1)*2^(2m-1)-1 and (6a-1)*2^(2m)-1, Math. Comp., 22 (1968), 420-422.
Index entries for sequences of n such that k*2^n-1 (or k*2^n+1) is prime

Cf. A002254 (5*2^n+1 is prime), A050522 (primes of the form 5*2^n - 1).

A001770:=n->`if`(isprime(5*2^n-1),n,NULL): seq(A001770(n), n=1..1000); # Wesley Ivan Hurt, Oct 15 2014

Select[Range[1000], PrimeQ[5*2^# - 1] &] (* Vaclav Kotesovec, Apr 28 2014 *)

is(n)=isprime(5*2^n-1) \\ Charles R Greathouse IV, Feb 07 2017

More terms from Hugo Pfoertner, Jun 23 2004

a(40) from the Wilfrid Keller link by Robert Price, Dec 22 2018

A268661 Numbers n such that 5*2^n + 1 is a prime factor of a generalized Fermat number 3^(2^m) + 1 for some m.

Original entry on oeis.org

3, 55, 127, 13165, 240937, 819739, 1282755

Offset: 1

Arkadiusz Wesolowski, Feb 10 2016

Wilfrid Keller, private communication, 2008.

Anders Björn and Hans Riesel, Factors of generalized Fermat numbers, Math. Comp. 67 (1998), no. 221, pp. 441-446.
Anders Björn and Hans Riesel, Table errata to “Factors of generalized Fermat numbers”, Math. Comp. 74 (2005), no. 252, p. 2099.
Anders Björn and Hans Riesel, Table errata 2 to "Factors of generalized Fermat numbers", Math. Comp. 80 (2011), pp. 1865-1866.
C. K. Caldwell, Top Twenty page, Generalized Fermat Divisors (base=3)
OEIS Wiki, Generalized Fermat numbers

Cf. A059919, A268662, A268663, A226366, A268664, A268657, A268658, A204620, A268659, A268660. Subsequence of A002254.

A268662 Numbers n such that 5*2^n + 1 is a prime factor of a generalized Fermat number 5^(2^m) + 1 for some m.

Original entry on oeis.org

7, 15, 25, 39, 55, 75, 85, 127, 1947, 3313, 13165, 23473, 125413, 1282755, 1777515

Offset: 1

Arkadiusz Wesolowski, Feb 10 2016

Wilfrid Keller, private communication, 2008.

Anders Björn and Hans Riesel, Factors of generalized Fermat numbers, Math. Comp. 67 (1998), no. 221, pp. 441-446.
Anders Björn and Hans Riesel, Table errata to “Factors of generalized Fermat numbers”, Math. Comp. 74 (2005), no. 252, p. 2099.
Anders Björn and Hans Riesel, Table errata 2 to "Factors of generalized Fermat numbers", Math. Comp. 80 (2011), pp. 1865-1866.
C. K. Caldwell, Top Twenty page, Generalized Fermat Divisors (base=5)
OEIS Wiki, Generalized Fermat numbers

Cf. A199591, A268661, A268663, A226366, A268664, A268657, A268658, A204620, A268659, A268660. Subsequence of A002254.

A268663 Numbers n such that 5*2^n + 1 is a prime factor of a generalized Fermat number 6^(2^m) + 1 for some m.

Original entry on oeis.org

127, 4687, 1777515

Offset: 1

Arkadiusz Wesolowski, Feb 10 2016

Wilfrid Keller, private communication, 2008.

Anders Björn and Hans Riesel, Factors of generalized Fermat numbers, Math. Comp. 67 (1998), no. 221, pp. 441-446.
Anders Björn and Hans Riesel, Table errata to “Factors of generalized Fermat numbers”, Math. Comp. 74 (2005), no. 252, p. 2099.
Anders Björn and Hans Riesel, Table errata 2 to "Factors of generalized Fermat numbers", Math. Comp. 80 (2011), pp. 1865-1866.
C. K. Caldwell, Top Twenty page, Generalized Fermat Divisors (base=6)
OEIS Wiki, Generalized Fermat numbers

Cf. A078303, A268661, A268662, A226366, A268664, A268657, A268658, A204620, A268659, A268660. Subsequence of A002254.

A268664 Numbers n such that 5*2^n + 1 is a prime factor of a generalized Fermat number 12^(2^m) + 1 for some m.

Original entry on oeis.org

13, 15, 127, 5947, 26607, 1320487

Offset: 1

Arkadiusz Wesolowski, Feb 10 2016

Wilfrid Keller, private communication, 2008.

Anders Björn and Hans Riesel, Factors of generalized Fermat numbers, Math. Comp. 67 (1998), no. 221, pp. 441-446.
Anders Björn and Hans Riesel, Table errata to “Factors of generalized Fermat numbers”, Math. Comp. 74 (2005), no. 252, p. 2099.
Anders Björn and Hans Riesel, Table errata 2 to "Factors of generalized Fermat numbers", Math. Comp. 80 (2011), pp. 1865-1866.
C. K. Caldwell, Top Twenty page, Generalized Fermat Divisors (base=12)
OEIS Wiki, Generalized Fermat numbers

Cf. A152585, A268661, A268662, A268663, A226366, A268657, A268658, A204620, A268659, A268660. Subsequence of A002254.

A050526 Primes of form 5*2^n+1.

Original entry on oeis.org

11, 41, 641, 40961, 163841, 167772161, 2748779069441, 180143985094819841, 188894659314785808547841, 193428131138340667952988161, 850705917302346158658436518579420528641

Offset: 1

N. J. A. Sloane, Dec 29 1999

nonn

All terms are odd since if n is even, then 5*2^n+1 is divisible by 3. - Michele Fabbrini, Jun 06 2021

Muniru A Asiru, Table of n, a(n) for n = 1..12
Ray Ballinger, Wilfried Keller, List of primes k*2^n + 1 for k < 300
Wikipedia, Proth's theorem
Index entries for sequences of n such that k*2^n-1 (or k*2^n+1) is prime

For the corresponding exponents n see A002254.

Cf. A019434, A039687, A050527, A050528, A050529, A195745, A300406, A300407, A300408.

Filtered(List([1..270], n->5*2^n + 1), IsPrime); # Muniru A Asiru, Mar 06 2018

[a: n in [1..200] | IsPrime(a) where a is 5*2^n + 1]; // Vincenzo Librandi, Mar 06 2018

a:=(n, k)->`if`(isprime(k*2^n+1), k*2^n+1, NULL):
seq(a(n, 5), n=1..127); # Martin Renner, Mar 05 2018

lista(nn) = {for(k=1, nn, if(ispseudoprime(p=5*2^k+1), print1(p, ", "))); } \\ Altug Alkan, Mar 29 2018

a(n) = A083575(A002254(n)). - Michel Marcus, Mar 29 2018

A112245 Numbers k such that 65537*2^k+1 is prime.

Original entry on oeis.org

287, 1695, 81359, 512895

Offset: 1

T. D. Noe, Aug 30 2005, Aug 26 2007

Note that 65537=2^16+1 is the largest known Fermat prime. These n yield provable primes. The primes are the smallest numbers in classes 303, 1711 and 81375 of the phi iteration (see A007755).

Jacques Molne found 512895. The corresponding provable prime is the smallest number in class 512911 of the Phi iteration.

Cf. A002253, A002254, A002259, A053345 (F*2^n+1 is prime, where F is a Fermat prime).

is(n)=ispseudoprime(65537*2^n+1) \\ Charles R Greathouse IV, Jun 13 2017

A282945 Numbers k such that 5*2^k + 1 is a prime factor of a generalized Fermat number 7^(2^m) + 1 for some m.

Original entry on oeis.org

15, 13165, 23473, 1777515

Offset: 1

Arkadiusz Wesolowski, Feb 25 2017

Anders Björn and Hans Riesel, Factors of generalized Fermat numbers, Math. Comp. 67 (1998), no. 221, pp. 441-446.
Anders Björn and Hans Riesel, Table errata to "Factors of generalized Fermat numbers", Math. Comp. 74 (2005), no. 252, p. 2099.
Anders Björn and Hans Riesel, Table errata 2 to "Factors of generalized Fermat numbers", Math. Comp. 80 (2011), pp. 1865-1866.
OEIS Wiki, Generalized Fermat numbers

Cf. A078304, A226366, A268661, A268662, A268663, A282946, A268664.

Subsequence of A002254.

lst = {}; Do[p = 5*2^n + 1; If[PrimeQ[p] && IntegerQ@Log[2, MultiplicativeOrder[7, p]], AppendTo[lst, n]], {n, 1, 13165, 2}]; lst

cp's OEIS Frontend

A002253 Numbers k such that 3*2^k + 1 is prime.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

PARI

PARI

Formula

Extensions

A226366 Numbers k such that 5*2^k + 1 is a prime factor of a Fermat number 2^(2^m) + 1 for some m.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A001770 Numbers k such that 5*2^k - 1 is prime.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

A268661 Numbers n such that 5*2^n + 1 is a prime factor of a generalized Fermat number 3^(2^m) + 1 for some m.

Views

Author

Keywords

References

Links

Crossrefs

A268662 Numbers n such that 5*2^n + 1 is a prime factor of a generalized Fermat number 5^(2^m) + 1 for some m.

Views

Author

Keywords

References

Links

Crossrefs

A268663 Numbers n such that 5*2^n + 1 is a prime factor of a generalized Fermat number 6^(2^m) + 1 for some m.

Views

Author

Keywords

References

Links

Crossrefs

A268664 Numbers n such that 5*2^n + 1 is a prime factor of a generalized Fermat number 12^(2^m) + 1 for some m.

Views

Author

Keywords

References

Links

Crossrefs

A050526 Primes of form 5*2^n+1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

PARI

Formula

A112245 Numbers k such that 65537*2^k+1 is prime.

Views