A070852 -id:A070852 - cp's OEIS frontend

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

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

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

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

A070848 Smallest prime == 1 mod (4n).

Original entry on oeis.org

5, 17, 13, 17, 41, 73, 29, 97, 37, 41, 89, 97, 53, 113, 61, 193, 137, 73, 229, 241, 337, 89, 277, 97, 101, 313, 109, 113, 233, 241, 373, 257, 397, 137, 281, 433, 149, 457, 157, 641, 821, 337, 173, 353, 181, 1289, 941, 193, 197, 401, 409, 1249, 1061, 433, 661

Offset: 1

Amarnath Murthy, May 15 2002

nonn

Note interesting patterns in the graph. - Zak Seidov, Dec 13 2011

			5 is the smallest prime of the form 1+4m, 17 is the smallest prime of the form 1+8m, 13 is the smallest prime of the form 1+12m, etc. - _Zak Seidov_, Dec 13 2011

Zak Seidov, Table of n, a(n) for n = 1..10000

Cf. A070846, A070847, A070849, A070850, A070851, A070852, A070853.

Cf. A034694.

nn=100; Reap[Do[p=1+4n; While[!PrimeQ[p], p=p+4n]; Sow[p], {n,nn}]][[2,1]] (* Zak Seidov, Dec 13 2011 *)

for(n=1,80,s=1; while((isprime(s)*s-1)%(4*n)>0,s++); print1(s,","))

nn=10000;for(n=1,nn,s=1+4*n;while(!isprime(s),s=s+4*n);print1(s,", ")) \\ Zak Seidov, Dec 13 2011

from sympy import isprime
def a(n):
  k = 4*n + 1
  while not isprime(k): k += 4*n
  return k
print([a(n) for n in range(1, 56)]) # Michael S. Branicky, May 17 2021

More terms from Benoit Cloitre, May 18 2002

A070849 Smallest prime == 1 mod (5n).

Original entry on oeis.org

11, 11, 31, 41, 101, 31, 71, 41, 181, 101, 331, 61, 131, 71, 151, 241, 1021, 181, 191, 101, 211, 331, 461, 241, 251, 131, 271, 281, 1451, 151, 311, 641, 331, 1021, 701, 181, 1481, 191, 1171, 401, 821, 211, 431, 661, 1801, 461, 941, 241, 491, 251, 1021, 521

Offset: 1

Amarnath Murthy, May 15 2002

nonn

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

Cf. A070846, A070847, A070848, A070850, A070851, A070852, A070853.

Cf. A034694.

for(n=1,80,s=1; while((isprime(s)*s-1)%(5*n)>0,s++); print1(s,","))

More terms from Benoit Cloitre, May 18 2002

A365958 Least k such that 8nk+1 is a prime.

Original entry on oeis.org

2, 1, 3, 3, 1, 2, 2, 3, 1, 3, 1, 1, 3, 1, 2, 2, 1, 3, 3, 4, 2, 2, 7, 1, 2, 6, 2, 2, 1, 1, 6, 1, 5, 5, 1, 2, 2, 4, 1, 2, 7, 1, 3, 1, 5, 9, 3, 2, 8, 1, 1, 3, 4, 1, 2, 1, 1, 2, 4, 7, 2, 3, 2, 15, 1, 4, 3, 10, 3, 5, 1, 1, 3, 1, 1, 2, 1, 2, 6, 1, 2, 12, 3, 1, 2

Offset: 1

Robert Price, Dec 17 2023

nonn

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

Cf. A016014, A005123.

A070852 lists the corresponding primes.

A365958 = {};
Do[k=1; While[!PrimeQ[8 n k+1], k++]; AppendTo[A365958 ,k], {n,85}];
A365958

a(n) = my(k=1); while (!isprime(8*n*k+1), k++); k; \\ Michel Marcus, Dec 17 2023

cp's OEIS Frontend

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

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

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A070848 Smallest prime == 1 mod (4n).

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A070849 Smallest prime == 1 mod (5n).

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A365958 Least k such that 8*n*k+1 is a prime.

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A365958 Least k such that 8nk+1 is a prime.