A070846 -id:A070846 - cp's OEIS frontend

A016014 Least k such that 2nk + 1 is a prime.

1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 3, 2, 1, 1, 3, 3, 1, 5, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 5, 3, 1, 2, 1, 1, 2, 3, 1, 3, 1, 4, 2, 1, 2, 3, 3, 1, 2, 1, 1, 3, 1, 1, 3, 1, 2, 2, 6, 2, 3, 3, 1, 2, 1, 3, 2, 1, 1, 2, 4, 3, 2, 1, 1, 3, 3, 1, 2, 4, 1, 5, 1, 2, 6, 1, 2, 2, 1, 1, 3, 7, 2, 5, 1, 1, 2, 1, 1

Offset: 1

Robert G. Wilson v

nonn

Is the sequence bounded? - Zak Seidov, Mar 25 2014
Answer: No, for any given N a number n such that a(n) > N can be constructed by the Chinese Remainder Theorem, see A239727. - Charles R Greathouse IV, Mar 25 2014
a(n) = 1 for n in A005097. - Robert Israel, Oct 26 2016

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

Cf. A005097, A103961.
A070846 contains the corresponding primes.
Records are in A239746 with indices in A239727.

f:= proc(n) local k;
     for k from 1 do if isprime(2*n*k+1) then return k fi od
end proc:
map(f, [$1..100]); # Robert Israel, Oct 26 2016

Do[k = 1; cp = n*k + 1; While[ ! PrimeQ[cp], k++; cp = n*k + 1]; Print[k], {n, 2, 400, 2}] (* Lei Zhou, Feb 23 2005 *)
lk[n_]:=Module[{k=1},While[!PrimeQ[2n k+1],k++];k]; Array[lk,100] (* Harvey P. Dale, Apr 23 2023 *)

a(n)=my(k); while(!isprime(2*n*(k++)+1),);k \\ Charles R Greathouse IV, Mar 25 2014

from sympy import isprime
def a(n):
    k = 1
    while not isprime(2*n*k + 1): k += 1
    return k
print([a(n) for n in range(1, 100)]) # Michael S. Branicky, Mar 28 2022

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

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

A070854 Smallest prime == 1 mod (10^n).

Original entry on oeis.org

11, 101, 3001, 70001, 700001, 22000001, 30000001, 600000001, 6000000001, 30000000001, 1900000000001, 18000000000001, 40000000000001, 3900000000000001, 6000000000000001, 130000000000000001, 3700000000000000001, 15000000000000000001, 150000000000000000001, 600000000000000000001, 16000000000000000000001

Offset: 1

Amarnath Murthy, May 15 2002

nonn

a(6) through a(21) have been certified prime with Primo. - Rick L. Shepherd, Jun 03 2002

Cf. A070846 to A070853.

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

a(n) = A121172(n)*10^n + 1. - Ray Chandler, Feb 10 2009

More terms from Rick L. Shepherd, Jun 03 2002

A341861 Number of primes among the (p-1)/2 numbers {2p+1, 4p+1, ..., (p-1)*p+1}, p = prime(n).

Original entry on oeis.org

1, 1, 2, 3, 4, 3, 2, 4, 4, 3, 4, 5, 7, 8, 5, 8, 7, 9, 9, 10, 11, 11, 12, 12, 14, 13, 13, 12, 15, 14, 14, 17, 15, 19, 18, 12, 19, 13, 19, 20, 22, 20, 24, 21, 15, 21, 21, 23, 25, 26, 23, 26, 26, 19, 23, 27, 24, 29, 27, 26, 28, 31, 29, 30, 25, 30, 30, 34, 31, 29, 35

Offset: 2

Jianing Song, Feb 21 2021

nonn

By Dirichlet's theorem on arithmetic progressions, we know there exists a prime q of the form 2*k*p+1. But the theorem does not give us any information about the size of the smallest q. It is conjectured that q < p^2. Moreover, it seems that a(n) goes to infinity as n increases.

			Let P denote the set of prime numbers. Then:
a(8) = #({39, 77, 115, 153, 191, 229, 267, 305, 343} intersect P) = #{191, 229} = 2.
a(11) = #({63, 125, 187, 249, 311, 373, 435, 497, 559, 621, 683, 745, 807, 869, 931} intersect P) = #{311, 373, 683} = 3.

Jianing Song, Table of n, a(n) for n = 2..10000

Cf. A070846.

a(n) = my(p=prime(n)); sum(k=1, (p-1)/2, isprime(2*k*p+1))

cp's OEIS Frontend

A016014 Least k such that 2*n*k + 1 is a prime.

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

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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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A016014 Least k such that 2nk + 1 is a prime.

A341861 Number of primes among the (p-1)/2 numbers {2p+1, 4p+1, ..., (p-1)*p+1}, p = prime(n).