A034694 -id:A034694 - cp's OEIS frontend

A038700 Smallest prime == -1 (mod n).

2, 3, 2, 3, 19, 5, 13, 7, 17, 19, 43, 11, 103, 13, 29, 31, 67, 17, 37, 19, 41, 43, 137, 23, 149, 103, 53, 83, 173, 29, 61, 31, 131, 67, 139, 71, 73, 37, 233, 79, 163, 41, 257, 43, 89, 137, 281, 47, 97, 149, 101, 103, 211, 53, 109, 167, 113, 173, 353, 59, 487, 61, 251

Offset: 1

David W. Wilson

Vladimir Pletser, Table of n, a(n) for n = 1..20000 (first 1000 terms from Joerg Arndt)

Cf. A034694.

f[n_] := Block[{k = n - 1}, While[ !PrimeQ@k, k += n]; k]; Array[f, 63] (* Robert G. Wilson v, Jun 09 2009 *)
With[{prs=Prime[Range[100]]},Table[SelectFirst[prs,Mod[#,n]==n-1&],{n,70}]] (* Harvey P. Dale, Apr 26 2023 *)

N=10^8;  default(primelimit,N);
a(n)=forprime(p=2,10^7, if(p%n==n-1,return(p)));
/* Joerg Arndt, Apr 16 2013 */

a(n)=my(j);while(!isprime(j++*n-1),);j*n-1 \\ Charles R Greathouse IV, Apr 18 2013

A070846 Smallest prime == 1 (mod 2n).

Original entry on oeis.org

3, 5, 7, 17, 11, 13, 29, 17, 19, 41, 23, 73, 53, 29, 31, 97, 103, 37, 191, 41, 43, 89, 47, 97, 101, 53, 109, 113, 59, 61, 311, 193, 67, 137, 71, 73, 149, 229, 79, 241, 83, 337, 173, 89, 181, 277, 283, 97, 197, 101, 103, 313, 107, 109, 331, 113, 229, 233, 709, 241

Offset: 1

Amarnath Murthy, May 15 2002

nonn

From Jianing Song, Feb 14 2021: (Start)
a(n) is the smallest prime p such that there is a primitive 2n-th root of unity modulo p, i.e., there is an element with order 2n in the multiplicative group of integers modulo p.
For n > 1, a(n) is the smallest prime p such that the 2n-th cyclotomic field Q(exp(2*Pi*i/(2*n))) can be embedded into the p-adic field Q_p. (End)

Dmitry Kamenetsky, Table of n, a(n) for n = 1..50000

Cf. A070847, A070848, A070849, A070850, A070851, A070852, A070853.

Cf. A034694, A016014.

With[{prs=Prime[Range[200]]},Flatten[Table[Select[prs,Mod[#,2n]==1&,1],{n,60}]]] (* Harvey P. Dale, Jan 16 2013 *)

for(n=1,80,s=1; while((isprime(s)*s-1)%(2*n)>0,s++); print1(s,","))

a(n) = 2*n*A016014(n) + 1. - Dmitry Kamenetsky, Oct 26 2016

More terms from Benoit Cloitre, May 18 2002

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

A050921 Smallest prime of form n*2^m+1, m >= 0, or 0 if no such prime exists.

Original entry on oeis.org

2, 3, 7, 5, 11, 7, 29, 17, 19, 11, 23, 13, 53, 29, 31, 17, 137, 19, 1217, 41, 43, 23, 47, 97, 101, 53, 109, 29, 59, 31, 7937, 257, 67, 137, 71, 37, 149, 1217, 79, 41, 83, 43, 173, 89, 181, 47

Offset: 1

N. J. A. Sloane, Dec 30 1999

Primes arising from A040076 (or 0 if no such prime exists).

Or: Starting with x=n+1, the first prime created by iterating the map x-> 2*x-1. - Kevin L. Schwartz and Christian N. K. Anderson, May 13 2013

R. J. Mathar, Table of n, a(n) for n = 1..382

Cf. A040076, A034694, A038699.

A050921 := proc(n)
    for m from 0 do
        if isprime(n*2^m+1) then
            return n*2^m+1 ;
        end if;
    end do;
end proc; # R. J. Mathar, Jun 01 2013

Do[m = 0; While[ !PrimeQ[n*2^m + 1], m++ ]; Print[n*2^m + 1], {n, 1, 47} ]

The next term (47*2^583 + 1) is too large to show.

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

A070853 Smallest prime == 1 mod (9n).

Original entry on oeis.org

19, 19, 109, 37, 181, 109, 127, 73, 163, 181, 199, 109, 937, 127, 271, 433, 307, 163, 2053, 181, 379, 199, 829, 433, 1801, 937, 487, 757, 523, 271, 1117, 577, 1783, 307, 631, 1297, 1999, 2053, 3511, 1801, 739, 379, 1549, 397, 811, 829, 1693, 433, 883

Offset: 1

Amarnath Murthy, May 15 2002

nonn

Robert Price, Table of n, a(n) for n = 1..5000

Cf. A070846, A070847, A070848, A070849, A070850, A070851, A070852.

Cf. A034694.

for(n=1,80,s=1; while((isprime(s)*s-1)%(9*n)>0,s++); print1(s,","))

Corrected and extended by Benoit Cloitre, May 18 2002

A070850 Smallest prime == 1 mod (6n).

Original entry on oeis.org

7, 13, 19, 73, 31, 37, 43, 97, 109, 61, 67, 73, 79, 337, 181, 97, 103, 109, 229, 241, 127, 397, 139, 433, 151, 157, 163, 337, 349, 181, 373, 193, 199, 409, 211, 433, 223, 229, 937, 241, 739, 757, 1033, 1321, 271, 277, 283, 577, 883, 601, 307, 313, 3181, 1297

Offset: 1

Amarnath Murthy, May 15 2002

nonn

Robert Price, Table of n, a(n) for n = 1..5000

Cf. A070846, A070847, A070848, A070849, A070851, A070852, A070853.

Cf. A034694.

for(n=1,80,s=1; while((isprime(s)*s-1)%(6*n)>0,s++); print1(s,","))

More terms from Benoit Cloitre, May 18 2002

A070851 Smallest prime == 1 mod (7n).

Original entry on oeis.org

29, 29, 43, 29, 71, 43, 197, 113, 127, 71, 463, 337, 547, 197, 211, 113, 239, 127, 1597, 281, 883, 463, 967, 337, 701, 547, 379, 197, 2437, 211, 1303, 449, 463, 239, 491, 757, 2591, 1597, 547, 281, 1723, 883, 3011, 617, 631, 967, 659, 337, 1373, 701, 1429

Offset: 1

Amarnath Murthy, May 15 2002

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..500

Cf. A070846 to A070850 and A070852, A070853.

Cf. A034694.

Module[{nn=60,prs=Prime[Range[500]]},Table[SelectFirst[prs,Mod[#,7n]==1&],{n,nn}]] (* Harvey P. Dale, Apr 13 2022 *)

for(n=1,80,s=1; while((isprime(s)*s-1)%(7*n)>0,s++); print1(s,","))

More terms from Benoit Cloitre, May 18 2002

A070852 Smallest prime == 1 mod (8n).

Original entry on oeis.org

17, 17, 73, 97, 41, 97, 113, 193, 73, 241, 89, 97, 313, 113, 241, 257, 137, 433, 457, 641, 337, 353, 1289, 193, 401, 1249, 433, 449, 233, 241, 1489, 257, 1321, 1361, 281, 577, 593, 1217, 313, 641, 2297, 337, 1033, 353, 1801, 3313, 1129, 769, 3137, 401

Offset: 1

Amarnath Murthy, May 15 2002

nonn

Robert Price, Table of n, a(n) for n = 1..5000

Cf. A070846 to A070851 and A070853.

Cf. A034694.

for(n=1,80,s=1; while((isprime(s)*s-1)%(8*n)>0,s++); print1(s,","))

More terms from Benoit Cloitre, May 18 2002

A070847 Smallest prime == 1 mod (3n).

Original entry on oeis.org

7, 7, 19, 13, 31, 19, 43, 73, 109, 31, 67, 37, 79, 43, 181, 97, 103, 109, 229, 61, 127, 67, 139, 73, 151, 79, 163, 337, 349, 181, 373, 97, 199, 103, 211, 109, 223, 229, 937, 241, 739, 127, 1033, 397, 271, 139, 283, 433, 883, 151, 307, 157, 3181, 163, 331, 337

Offset: 1

Amarnath Murthy, May 15 2002

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A070846, A070848, A070849, A070850, A070851, A070852, A070853.
Cf. A034694.
Cf. A024892 (n such that a(n)=3*n+1).
Cf. A002476.

f:= proc(n) local k,d;
  if n::even then d:= 3*n else d:= 6*n fi;
  for k from 1 by d do if isprime(k) then return k fi od
end proc:
map(f, [$1..100]); # Robert Israel, Sep 19 2019

a[n_] := Module[{k, d}, If[EvenQ[n], d = 3n, d = 6n]; For[k = 1, True, k += d, If[PrimeQ[k], Return[k]]]];
Array[a, 100] (* Jean-François Alcover, Jun 11 2020, after Maple *)

for(n=1,80,s=1; while((isprime(s)*s-1)%(3*n)>0,s++); print1(s,","))

More terms from Benoit Cloitre, May 18 2002

cp's OEIS Frontend

A038700 Smallest prime == -1 (mod n).

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A070846 Smallest prime == 1 (mod 2n).

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

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A050921 Smallest prime of form n*2^m+1, m >= 0, or 0 if no such prime exists.

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Maple

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A141849 Primes congruent to 1 mod 11.

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A070853 Smallest prime == 1 mod (9n).

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A070850 Smallest prime == 1 mod (6n).

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A070851 Smallest prime == 1 mod (7n).

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