A007522 -id:A007522 - cp's OEIS frontend

A014663 Primes p such that multiplicative order of 2 modulo p is odd.

7, 23, 31, 47, 71, 73, 79, 89, 103, 127, 151, 167, 191, 199, 223, 233, 239, 263, 271, 311, 337, 359, 367, 383, 431, 439, 463, 479, 487, 503, 599, 601, 607, 631, 647, 719, 727, 743, 751, 823, 839, 863, 881, 887, 911, 919, 937, 967, 983, 991, 1031, 1039, 1063

Offset: 1

N. J. A. Sloane, Dec 11 1999

nonn

Or, primes p which do not divide 2^n+1 for any n.
The possibility n=0 in the above rules out A072936(1)=2; apart from this, a(n)=A072936(n+1). - M. F. Hasler, Dec 08 2007
The order of 2 mod p is odd iff 2^k=1 mod p, where p-1=2^s*k, k odd. - M. F. Hasler, Dec 08 2007
Has density 7/24 (Hasse).
From Jianing Song, Jun 27 2025: (Start)
The multiplicative order of 2 modulo a(n) is A139686(n).
Contained in primes congruent to 1 or 7 modulo 8 (primes p such that 2 is a quadratic residue modulo p, A001132), and contains primes congruent to 7 modulo 8 (A007522). (End)

Christopher Adler and Jean-Paul Allouche (2022), Finite self-similar sequences, permutation cycles, and music composition, Journal of Mathematics and the Arts, 16:3, 244-261, DOI: 10.1080/17513472.2022.2116745.
P. Moree, Appendix to V. Pless et al., Cyclic Self-Dual Z_4 Codes, Finite Fields Applic., vol. 3 pp. 48-69, 1997.

T. D. Noe, Table of n, a(n) for n=1..1000
H. H. Hasse, Über die Dichte der Primzahlen p, ... , Math. Ann., 168 (1966), 19-23.
J. C. Lagarias, The set of primes dividing the Lucas numbers has density 2/3, Pacific J. Math., 118. No. 2, (1985), 449-461.
Chunlei Li, Nian Li, and Matthew G. Parker, Complementary Sequence Pairs of Types II and III. [From _N. J. A. Sloane_, Jun 16 2012]

Cf. Complement in primes of A091317.
Cf. A001132, A007522, A040098, A045315, A049564, A139686 (the actual multiplicative orders).
Cf. Essentially the same as A072936 (except for missing leading term 2).
Cf. other bases: this sequence (base 2), A385220 (base 3), A385221 (base 4), A385192 (base 5), A163183 (base -2), A385223 (base -3), A385224 (base -4), A385225 (base -5).

okQ[p_] := OddQ[MultiplicativeOrder[2, p]];
Select[Prime[Range[1000]], okQ] (* Jean-François Alcover, Nov 23 2024 *)

isA014663(p)=1==Mod(1,p)<<((p-1)>>factor(p-1,2)[1,2])
listA014663(N=1000)=forprime(p=3,N,isA014663(p)&print1(p", ")) \\ M. F. Hasler, Dec 08 2007

lista(nn) = {forprime(p=3, nn, if (znorder(Mod(2, p)) % 2, print1(p, ", ")););} \\ Michel Marcus, Feb 06 2015

Edited by M. F. Hasler, Dec 08 2007

More terms from Max Alekseyev, Feb 06 2010

A005122 Numbers k such that 8k - 1 is prime.

Original entry on oeis.org

1, 3, 4, 6, 9, 10, 13, 16, 19, 21, 24, 25, 28, 30, 33, 34, 39, 45, 46, 48, 54, 55, 58, 60, 61, 63, 75, 76, 79, 81, 90, 91, 93, 94, 103, 105, 108, 111, 114, 115, 121, 123, 124, 129, 130, 133, 136, 138, 144, 153, 154, 160, 163, 165, 166, 171, 175, 178, 180

Offset: 1

N. J. A. Sloane

Corresponding primes are listed in A007522. - Altug Alkan, Dec 10 2015

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A007522.

[n: n in [1..200] | IsPrime(8*n-1)]; // Vincenzo Librandi, Dec 10 2015

A005122:=n->`if`(isprime(8*n-1), n, NULL): seq(A005122(n), n=1..300); # Wesley Ivan Hurt, Dec 11 2015

Select[Range[160], PrimeQ[8#-1]&] (* Harvey P. Dale, Nov 25 2011 *)

isok(n) = isprime(8*n-1); \\ Michel Marcus, Dec 10 2015

A127581 Smallest prime of the form k*2^n - 1, for k >= 2.

Original entry on oeis.org

2, 3, 7, 23, 31, 127, 127, 383, 1279, 3583, 5119, 6143, 8191, 73727, 81919, 131071, 131071, 524287, 524287, 14680063, 14680063, 14680063, 109051903, 109051903, 654311423, 738197503, 738197503, 2147483647, 2147483647, 2147483647

Offset: 0

Artur Jasinski, Jan 19 2007

nonn

			a(3)=23 because 23 = 3*2^3 - 1 is prime.
a(4)=31 because 31 = 2*2^4 - 1 is prime.

Cf. A007522, A127575-A127582, A127586-A127587.

a = {}; Do[k = 1; While[ !PrimeQ[k 2^n + 2^n - 1], k++ ]; AppendTo[a, k 2^n + 2^n - 1], {n, 0, 50}]; a

a(n) << 37^n by Xylouris' improvement to Linnik's theorem. - Charles R Greathouse IV, Dec 10 2013

Edited by Don Reble, Jun 11 2007

A127586 Smallest strictly positive integer k such that (k+1)*2^n-1 is prime.

Original entry on oeis.org

2, 1, 1, 2, 1, 3, 1, 2, 4, 6, 4, 2, 1, 8, 4, 3, 1, 3, 1, 27, 13, 6, 25, 12, 38, 21, 10, 15, 7, 3, 1, 9, 4, 5, 2, 23, 11, 5, 2, 24, 23, 11, 5, 2, 13, 6, 19, 9, 4, 18, 10

Offset: 0

Artur Jasinski, Jan 19 2007

nonn

The associated prime number list is (k+1)*2^n-1 = 2, 3, 7, 23, 31, 127, 127, 383, 1279, 3583, 5119, 6143, 8191, 73727 for n=0,1,2,3,4,... - R. J. Mathar, Jan 22 2007

Cf. A007522, A127575, A127576, A127577, A127578, A127580, A127581, A087522, A127587.

A127586 := proc(n) local k; k:=1 ; while true do if isprime( (k+1)*2^n-1) then RETURN(k) ; fi ; k := k+1 ; od ; end: for n from 0 to 100 do printf("%d, ",A127586(n)) ; od ; # R. J. Mathar, Jan 22 2007

a = {}; Do[k = 1; While[ ! PrimeQ[k 2^n + 2^n - 1], k++ ]; AppendTo[a, k], {n, 0, 50}]; a

a(n)=A127587(n) if n is not in A000043. - R. J. Mathar, Jan 22 2007

a(n) << 19^n by Xylouris' improvement to Linnik's theorem. - Charles R Greathouse IV, Dec 10 2013

A127587 Smallest nonnegative integer k such that (k+1)*2^n-1 is prime.

Original entry on oeis.org

2, 1, 0, 0, 1, 0, 1, 0, 4, 6, 4, 2, 1, 0, 4, 3, 1, 0, 1, 0, 13, 6, 25, 12, 38, 21, 10, 15, 7, 3, 1, 0, 4, 5, 2, 23, 11, 5, 2, 24, 23, 11, 5, 2, 13, 6, 19, 9, 4, 18, 10, 20, 19, 9, 4, 2, 31, 15, 7, 3, 1, 0, 11, 5, 2, 66, 62, 42, 62, 39, 19, 9, 4, 14, 11, 5, 2, 54, 46, 29, 14, 29, 14, 63, 31, 15, 7

Offset: 0

Artur Jasinski, Jan 19 2007

nonn

The associated prime number list is (k+1)*2^n-1 = 2,3,3,7,31,31,127,127,1279,3583,5119,6143,... for n=0,1,2,3,4,... - R. J. Mathar, Jan 22 2007

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A007522, A127575, A127576, A127577, A127578, A127580, A127581, A087522, A127586.

A127587 := proc(n) local k; k:=0 ; while true do if isprime( (k+1)*2^n-1) then RETURN(k) ; fi ; k := k+1 ; od ; end: for n from 0 to 100 do printf("%d, ",A127587(n)) ; od ; # R. J. Mathar, Jan 22 2007

a = {}; Do[k = 0; While[ ! PrimeQ[k 2^n + 2^n - 1], k++ ]; AppendTo[a, k], {n, 0, 50}]; a

a[A000043(j)]=0 for j=1,2,3,4,... - R. J. Mathar, Jan 22 2007

a(n) = A085427(n) - 1. - Filip Zaludek, Dec 16 2016

More terms from R. J. Mathar, Jan 22 2007

A127589 Primes of the form 16k + 5.

Original entry on oeis.org

5, 37, 53, 101, 149, 181, 197, 229, 277, 293, 373, 389, 421, 613, 661, 677, 709, 757, 773, 821, 853, 997, 1013, 1061, 1093, 1109, 1237, 1301, 1381, 1429, 1493, 1621, 1637, 1669, 1733, 1861, 1877, 1973, 2053, 2069, 2213, 2293, 2309, 2341, 2357, 2389, 2437

Offset: 1

Artur Jasinski, Jan 19 2007

All terms are the sum of two squares.

Primes with least significant digit 5 in hexadecimal. - Alonso del Arte, Oct 21 2022

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A035050, A007522, A127575, A127576, A127577, A127578, A127580, A127581, A087522, A127586, A127587.

a = {}; Do[If[PrimeQ[16n + 5], AppendTo[a, 16n + 5]], {n, 0, 200}]; a
Select[16Range[200] + 5, PrimeQ] (* Alonso del Arte, Oct 21 2022 *)

select(x->(x%16)==5, primes(500)) \\ Michel Marcus, Oct 24 2022

Invalid comment removed by Zak Seidov, Jul 22 2010

A035089 Smallest prime of form 2^n*k + 1.

Original entry on oeis.org

2, 3, 5, 17, 17, 97, 193, 257, 257, 7681, 12289, 12289, 12289, 40961, 65537, 65537, 65537, 786433, 786433, 5767169, 7340033, 23068673, 104857601, 167772161, 167772161, 167772161, 469762049, 2013265921, 3221225473, 3221225473, 3221225473, 75161927681

Offset: 0

Labos Elemer

nonn

a(n) is the smallest prime p such that the multiplicative group modulo p has a subgroup of order 2^n. - Joerg Arndt, Oct 18 2020

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Gareth A. Jones and Alexander K. Zvonkin, Groups of prime degree and the Bateman-Horn conjecture, 2021.

Analogous case is A034694. Fermat primes (A019434) are a subset. See also Fermat numbers A000215.

Cf. A007522, A057775, A127575, A127576, A127577, A127578, A127580, A127581, A087522, A127586, A127587.

a = {}; Do[k = 0; While[ !PrimeQ[k 2^n + 1], k++ ]; AppendTo[a, k 2^n + 1], {n, 1, 50}]; a (* Artur Jasinski *)

a(n)=for(k=1,9e99,if(ispseudoprime(k<Charles R Greathouse IV, Jul 06 2011

a(0) from Joerg Arndt, Jul 06 2011

A139487 Numbers k such that 8k + 7 is prime.

Original entry on oeis.org

0, 2, 3, 5, 8, 9, 12, 15, 18, 20, 23, 24, 27, 29, 32, 33, 38, 44, 45, 47, 53, 54, 57, 59, 60, 62, 74, 75, 78, 80, 89, 90, 92, 93, 102, 104, 107, 110, 113, 114, 120, 122, 123, 128, 129, 132, 135, 137, 143, 152, 153, 159, 162, 164, 165, 170, 174, 177, 179, 180, 183, 185

Offset: 1

Artur Jasinski, Apr 23 2008

For numbers k such that:
8k+1 is prime see A005123, primes see A007519;
8k+3 is prime see A005124, primes see A007520;
8k+5 is prime see A105133, primes see A007521;
8k+7 is prime see A139487, primes see A007522.
8k + 7 divides A000225(4k+3). - Jinyuan Wang, Mar 08 2019

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000225, A007519, A007520, A007521, A007522, A005123, A005124, A105133.

[n: n in [0..200] | IsPrime(8*n+7)]; // Vincenzo Librandi, Jun 25 2014

a = {}; Do[If[PrimeQ[8 n + 7], AppendTo[a, n]], {n, 0, 300}]; a
Select[Range[0,200],PrimeQ[8#+7]&] (* Harvey P. Dale, Oct 10 2012 *)

is(n)=isprime(8*n+7) \\ Charles R Greathouse IV, Feb 17 2017

a(n) = (A007522(n) - 7)/8, n >= 1.

A141164 Numbers having exactly 1 divisor of the form 8*k + 7.

Original entry on oeis.org

7, 14, 15, 21, 23, 28, 30, 31, 35, 39, 42, 45, 46, 47, 49, 55, 56, 60, 62, 69, 70, 71, 75, 77, 78, 79, 84, 87, 90, 91, 92, 93, 94, 95, 98, 103, 110, 111, 112, 115, 117, 120, 124, 127, 133, 138, 140, 141, 142, 143, 147, 150, 151, 154, 155, 156, 158, 159, 167, 168, 174, 180, 182, 183, 184, 186, 188, 190, 191, 196, 199

Offset: 1

Reinhard Zumkeller, Mar 26 2011

nonn

			a(1) = A188226(1) = 7.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Numbers having m divisors of the form 8*k + i: A343107 (m=1, i=1), A343108 (m=0, i=3), A343109 (m=0, i=5), A343110 (m=0, i=7), A343111 (m=2, i=1), A343112 (m=1, i=3), A343113 (m=1, i=5), this sequence (m=1, i=7).
Indices of 1 in A188172.
A007522 is a subsequence.
Cf. A004771.

import Data.List (elemIndices)
a141164 n = a141164_list !! (n-1)
a141164_list = map succ $ elemIndices 1 $ map a188172 [1..]

okQ[n_] := Length[Select[Divisors[n] - 7, Mod[#, 8] == 0 &]] == 1; Select[Range[200], okQ]

res(n, a, b) = sumdiv(n, d, (d%a) == b)
isA141164(n) = (res(n, 8, 7) == 1) \\ Jianing Song, Apr 06 2021

A188172(a(n)) = 1.

A141849 Primes congruent to 1 mod 11.

Original entry on oeis.org

23, 67, 89, 199, 331, 353, 397, 419, 463, 617, 661, 683, 727, 859, 881, 947, 991, 1013, 1123, 1277, 1321, 1409, 1453, 1607, 1783, 1871, 2003, 2069, 2113, 2179, 2267, 2311, 2333, 2377, 2399, 2531, 2663, 2707, 2729, 2861, 2927, 2971, 3037, 3169, 3191, 3257

Offset: 1

N. J. A. Sloane, Jul 11 2008

Conjecture: Also primes p such that ((x+1)^11-1)/x has 10 distinct irreducible factors of degree 1 over GF(p). - Federico Provvedi, Apr 17 2018

Primes congruent to 1 mod 22. - Chai Wah Wu, Apr 28 2025

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000040, A090187, A102656.
Prime sequences A# (k,r) of the form k*n+r with 0 <= r <= k-1 (i.e., primes == r (mod k), or primes p with p mod k = r) and gcd(r,k)=1: A000040 (1,0), A065091 (2,1), A002476 (3,1), A003627 (3,2), A002144 (4,1), A002145 (4,3), A030430 (5,1), A045380 (5,2), A030431 (5,3), A030433 (5,4), A002476 (6,1), A007528 (6,5), A140444 (7,1), A045392 (7,2), A045437 (7,3), A045471 (7,4), A045458 (7,5), A045473 (7,6), A007519 (8,1), A007520 (8,3), A007521 (8,5), A007522 (8,7), A061237 (9,1), A061238 (9,2), A061239 (9,4), A061240 (9,5), A061241 (9,7), A061242 (9,8), A030430 (10,1), A030431 (10,3), A030432 (10,7), A030433 (10,9), this sequence (11,1), A090187 (11,2), A141850 (11,3), A141851 (11,4), A141852 (11,5), A141853 (11,6), A141854 (11,7), A141855 (11,8), A141856 (11,9), A141857 (11,10), A068228 (12,1), A040117 (12,5), A068229 (12,7), A068231 (12,11).
Cf. A034694 (smallest prime == 1 (mod n)).
Cf. A038700 (smallest prime == n-1 (mod n)).
Cf. A038026 (largest possible value of smallest prime == r (mod n)).

Filtered([1..4000],n->n mod 11=1 and IsPrime(n)); # Muniru A Asiru, Apr 19 2018

[ p: p in PrimesUpTo(5000) | p mod 11 eq 1 ]; // Vincenzo Librandi, Apr 19 2011

a:=select(n->isprime(n) and modp(n,11)=1,[$1..4000]); # Muniru A Asiru, Apr 19 2018

Select[Range[1,10000,11],PrimeQ] (* Vladimir Joseph Stephan Orlovsky, May 18 2011 *)

is(n)=isprime(n) && n%11==1 \\ Charles R Greathouse IV, Jul 01 2016

forstep(n=2, 1e3, 2, if(isprime(p=11*n+1), print1(p, ", "))); \\ Altug Alkan, Apr 19 2018

a(n) ~ 10n log n. - Charles R Greathouse IV, Jul 02 2016

cp's OEIS Frontend

A014663 Primes p such that multiplicative order of 2 modulo p is odd.

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A005122 Numbers k such that 8k - 1 is prime.

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A127581 Smallest prime of the form k*2^n - 1, for k >= 2.

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A127586 Smallest strictly positive integer k such that (k+1)*2^n-1 is prime.

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Maple

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A127587 Smallest nonnegative integer k such that (k+1)*2^n-1 is prime.

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Maple

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A127589 Primes of the form 16k + 5.

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A035089 Smallest prime of form 2^n*k + 1.

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