A070847 -id:A070847 - 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

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

A366931 Least k such that 3nk+1 is a prime.

Original entry on oeis.org

2, 1, 2, 1, 2, 1, 2, 3, 4, 1, 2, 1, 2, 1, 4, 2, 2, 2, 4, 1, 2, 1, 2, 1, 2, 1, 2, 4, 4, 2, 4, 1, 2, 1, 2, 1, 2, 2, 8, 2, 6, 1, 8, 3, 2, 1, 2, 3, 6, 1, 2, 1, 20, 1, 2, 2, 12, 2, 4, 1, 2, 2, 2, 1, 6, 1, 8, 2, 4, 1, 4, 2, 2, 1, 8, 1, 2, 4, 6, 1, 2, 3, 2, 3, 4, 4

Offset: 1

Robert Price, Dec 17 2023

nonn

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

Cf. A016014, A024892.

A070847 lists the corresponding primes.

A366931 = {};
Do[k=1; While[!PrimeQ[3 n k+1], k++]; AppendTo[A366931 ,k], {n,85}];
A366931
lkp[n_]:=Module[{k=1},While[!PrimeQ[3n*k+1],k++];k]; Array[lkp,90] (* Harvey P. Dale, Jan 09 2025 *)

a(n) = my(k=1); while (!isprime(3*n*k+1), k++); k; \\ Michel Marcus, Dec 17 2023

A379150 Smallest prime ending in "3", with n preceding "0" digits.

Original entry on oeis.org

103, 2003, 70003, 100003, 1000003, 20000003, 500000003, 40000000003, 40000000003, 100000000003, 2000000000003, 230000000000003, 3100000000000003, 11000000000000003, 20000000000000003, 100000000000000003, 1000000000000000003, 310000000000000000003, 500000000000000000003

Offset: 1

James S. DeArmon, Dec 16 2024

Leading zeros are not allowed, e.g., "03".

a(997) has 1001 digits. - Michael S. Branicky, Dec 16 2024

			a(1) = 103, is the smallest prime ending in "03";
a(2) = 2003, is the smallest prime ending in "003".

Michael S. Branicky, Table of n, a(n) for n = 1..996

Cf. A070854, A070847.

Table[i=1;While[!PrimeQ[m=FromDigits[Join[IntegerDigits[i],Table[0,n],{3}]]],i++];m,{n,19}] (* James C. McMahon, Dec 23 2024 *)

a(n)=for(i=1, oo, if(isprime(i*10^(n+1)+3), return(i*10^(n+1)+3))) \\ Johann Peters, Dec 27 2024

import sympy
def prime3_finder():
  outVec = []
  power = 2
  for n in range(100,999999999):
      if not n & 3 == 3: continue # speed-up over simple MOD operation
      if not n % 10**power == 3: continue
      if not sympy.isprime(n): continue
      outVec.append(n)
      power += 1
  return outVec
outvec = prime3_finder()
print(outvec)

from sympy import isprime
from itertools import count
def a(n): return next(i for i in count(10**(n+1)+3, 10**(n+1)) if isprime(i))
print([a(n) for n in range(1, 20)]) # Michael S. Branicky, Dec 16 2024

More terms from Michael S. Branicky, Dec 16 2024

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

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

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A366931 Least k such that 3*n*k+1 is a prime.

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Mathematica

PARI

A379150 Smallest prime ending in "3", with n preceding "0" digits.

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Author

Keywords

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Programs

Mathematica

PARI

Python

Python

Extensions

A366931 Least k such that 3nk+1 is a prime.