A034694 -id:A034694 - cp's OEIS frontend

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

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

A085420 For each n, let p(n,b) be the smallest prime in the arithmetic progression k*n+b, with k > 0. Then a(n) = max(p(n,b)) with 0 < b < n and gcd(b,n) = 1.

Original entry on oeis.org

2, 3, 7, 7, 19, 11, 29, 23, 43, 19, 71, 23, 103, 53, 43, 43, 103, 53, 191, 59, 97, 79, 233, 73, 269, 103, 173, 83, 317, 79, 577, 151, 227, 193, 239, 157, 439, 191, 233, 157, 587, 107, 467, 257, 389, 307, 967, 191, 613, 269, 421, 601, 659, 199, 353, 233, 433, 317, 709

Offset: 1

T. D. Noe, Jun 29 2003

nonn

Linnik proved that there are n0 and L such that a(n) < n^L for all n > n0. It has been conjectured that a(n) < n^2. The sequence A034694 has the primes p(n,1).

a(n) is also the maximum term in row n of A060940, which defines a(1). A007918(n+1) is the minimum term in row n of A060940. - Seiichi Manyama, Apr 02 2018

			a(5) = 19 because p(5,1) = 11, p(5,2) = 7, p(5,3) = 13 and p(5,4) = 19.

P. Ribenboim, The New Book of Prime Number Records, Springer, 1996, pp. 277-284.

T. D. Noe, Table of n, a(n) for n = 1..10000 (corrected by Michel Marcus, Jan 19 2019)
A. Granville, Least primes in arithmetic progressions, Théorie des nombres / Number Theory (ed. J. M. De Koninck & C. Levesque), (de Gruyter, New York, 1989), 306-321.
Eric Weisstein's World of Mathematics, Linnik's Theorem

A038026 is a lower bound.

Cf. A034694.

minP[n_, a_] := Module[{k, p}, If[GCD[n, a]>1, p=0, k=1; While[ !PrimeQ[k*n+a], k++ ]; p=k*n+a]; p]; Table[Max[Table[minP[n, i], {i, n-1}]], {n, 2, 100}]

p(n,b)=while(!isprime(b+=n),); ba(n)=my(t=p(n,1));for(b=2,n-1,if(gcd(n,b)==1,t=max(t,p(n,b))));t \\ Charles R Greathouse IV, Sep 08 2012

a(1) defined via A060940 by Seiichi Manyama, Apr 02 2018

A064632 Smallest prime p such that n = (p-1)/(q-1) for some prime q.

Original entry on oeis.org

3, 7, 5, 11, 7, 29, 17, 19, 11, 23, 13, 53, 29, 31, 17, 103, 19, 191, 41, 43, 23, 47, 97, 101, 53, 109, 29, 59, 31, 311, 193, 67, 137, 71, 37, 149, 229, 79, 41, 83, 43, 173, 89, 181, 47, 283, 97, 197, 101, 103, 53, 107, 109, 331, 113, 229, 59, 709, 61, 367, 373

Offset: 2

Robert G. Wilson v, Oct 16 2001

			a(7) = 29 because (29-1)/(5-1).

Daria Micovic, Table of n, a(n) for n = 2..10000
Matthew M. Conroy, A sequence related to a conjecture of Schinzel, , J. Integ. Seqs. Vol. 4 (2001), #01.1.7.

Similar to but not the same as A034694. Cf. A064652 (q-values), A064673.

NextPrim[n_] := (k = n + 1; While[ !PrimeQ[k], k++ ]; k); Do[p = 2; While[q = (p - 1)/n + 1; !PrimeQ[q] || q >= p, p = NextPrim[p]]; Print[p], {n, 2, 100} ]
spp[n_]:=Module[{p=2},While[!PrimeQ[(p-1)/n+1],p=NextPrime[p]];p]; Array[ spp,70,2] (* Harvey P. Dale, Aug 22 2019 *)

a(n) = {forprime(p=2, , forprime(q=2, p-1, if ((p-1)/(q-1) == n, return (p));););} \\ Michel Marcus, Apr 16 2016

def A064632(n):
    p, q = 0, 0
    while not (q.is_prime() and q < p):
        p = next_prime(p)
        if p % n != 1: continue
        q = (p - 1) // n + 1
    return p # Daria Micovic, Apr 13 2016

Definition corrected by Stephanie Anderson, Apr 16 2016

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

A077316 Triangle read by rows: T(n,k) is the k-th prime = 1 (mod n).

Original entry on oeis.org

2, 3, 5, 7, 13, 19, 5, 13, 17, 29, 11, 31, 41, 61, 71, 7, 13, 19, 31, 37, 43, 29, 43, 71, 113, 127, 197, 211, 17, 41, 73, 89, 97, 113, 137, 193, 19, 37, 73, 109, 127, 163, 181, 199, 271, 11, 31, 41, 61, 71, 101, 131, 151, 181, 191, 23, 67, 89, 199, 331, 353

Offset: 1

Amarnath Murthy, Nov 04 2002

			Triangle begins:
   2;
   3,  5;
   7, 13, 19;
   5, 13, 17, 29;
  11, 31, 41, 61, 71;
  ...

Nathaniel Johnston, Rows 1..100, flattened

Cf. A034694 (first column), A077317 (main diagonal), A077318 (row sums), A077319, A093870, A193869 (row products).

Tj := proc(n,k) option remember: local j,p: if(k=0)then return 0:fi: for j from procname(n,k-1)+1 do if(isprime(n*j+1))then return j: fi: od: end: A077316 := proc(n,k) return n*Tj(n,k)+1: end: seq(seq(A077316(n,k),k=1..n),n=1..15); # Nathaniel Johnston, Sep 02 2011

Tj[n_, k_] := Tj[n, k] = Module[{j}, If[k == 0, Return[0]];
   For[j = Tj[n, k-1]+1, True, j++, If[PrimeQ[n*j+1], Return[j]]]];
T[n_, k_] := n*Tj[n, k]+1;
Table[Table[T[n, k], {k, 1, n}], {n, 1, 15}] // Flatten (* Jean-François Alcover, Aug 02 2022, after Nathaniel Johnston *)

Edited and extended by Franklin T. Adams-Watters, Aug 29 2006

A034782 Numbers n such that A034693(n) = 3: 3n + 1 is prime, but neither n + 1 nor 2n + 1.

Original entry on oeis.org

24, 32, 34, 64, 76, 80, 92, 94, 104, 110, 122, 124, 132, 144, 152, 154, 182, 202, 206, 214, 220, 236, 242, 244, 246, 252, 274, 286, 294, 302, 322, 332, 340, 344, 356, 362, 364, 374, 376, 390, 412, 416, 434, 474, 482, 484, 494, 496

Offset: 1

Labos Elemer

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A034693, A034694.

Position[#, 3] &@ Table[k = 1; While[! PrimeQ[k n + 1], k++]; k, {n, 500}] // Flatten (* Michael De Vlieger, Mar 02 2017 *)

is(n)=!isprime(n+1)&&!isprime(2*n+1)&&isprime(3*n+1) \\ M. F. Hasler, May 13 2018

Corrected by R. J. Mathar, Jul 26 2015

Name edited by M. F. Hasler, May 13 2018

A034784 Numbers n such that A034693(n) = 2.

Original entry on oeis.org

3, 5, 8, 9, 11, 14, 15, 20, 21, 23, 26, 29, 33, 35, 39, 41, 44, 48, 50, 51, 53, 54, 56, 63, 65, 68, 69, 74, 75, 81, 83, 86, 89, 90, 95, 98, 99, 105, 111, 113, 114, 116, 119, 120, 125, 128, 131, 134, 135, 140, 141, 146, 153, 155, 158, 165, 168, 173, 174, 176

Offset: 1

Labos Elemer

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A034693, A034694.

CompositeNext[n_]:=Module[{k=n+1},While[PrimeQ[k],k++ ];k]; lst={};Do[p=n+CompositeNext[n];If[PrimeQ[p],AppendTo[lst,n]],{n,2,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Jun 15 2009 *)
Position[#, 2] &@ Table[k = 1; While[! PrimeQ[k n + 1], k++]; k, {n, 180}] // Flatten (* Michael De Vlieger, Mar 02 2017 *)

Edited by N. J. A. Sloane, Oct 27 2012

A077317 a(n) is the n-th prime == 1 (mod n).

Original entry on oeis.org

2, 5, 19, 29, 71, 43, 211, 193, 271, 191, 661, 277, 937, 463, 691, 769, 1531, 613, 2357, 1021, 1723, 1409, 3313, 1609, 3701, 2029, 3187, 2437, 6961, 1741, 7193, 3617, 4951, 3877, 7001, 3169, 10657, 6271, 7879, 5521, 13613, 3823, 15137, 7349, 9091, 7499

Offset: 1

Amarnath Murthy, Nov 04 2002, Jul 23 2005 (submitted incorrectly on two separate occasions; each time it has had to be corrected)

nonn

a(n) is the n-th prime in the arithmetic progression with first term 1 and common difference n.

Main diagonal of A077316.

			a(5) = 71, the fifth prime in the sequence 1, 6, 11, 16, 21, 26, 31, 36, 41, 46, 51, 56, 61, 66, 71, ...

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A034694, A077316, A077318, A077319, A093871.

a:=proc(n) local N, B, j: N:=[seq(n*x+1,x=0..500)]: B:={}: for j from 1 to nops(N) do if isprime(N[j])=true then B:=B union {N[j]} else fi od: B[n]: end: seq(a(n),n=1..55); # Emeric Deutsch, Sep 03 2005

Module[{nn=50,prs=Prime[Range[3000]]},Join[{2},Table[Select[prs,Mod[#,n] == 1&][[n]],{n,2,nn}]]] (* Harvey P. Dale, Nov 13 2020 *)

Edited, corrected and extended by Emeric Deutsch, Sep 03 2005 and by Franklin T. Adams-Watters, Aug 29 2006

Edited by N. J. A. Sloane, Aug 23 2008 at the suggestion of R. J. Mathar

A077318 Sum of terms in n-th row of A077316.

Original entry on oeis.org

2, 8, 39, 64, 215, 150, 791, 760, 1179, 970, 3619, 1668, 6227, 3360, 5595, 6144, 13821, 5436, 24985, 10280, 17115, 14850, 35765, 18600, 42425, 26390, 43389, 34580, 92481, 24180, 115909, 57440, 79431, 61200, 115605, 55332, 197913, 106020

Offset: 1

Amarnath Murthy, Nov 04 2002

nonn

By definition a(n) == 0 (mod n).

It appears that a(n) is also the smallest sum of n distinct primes of the form n*k+1. - Omar E. Pol, Sep 03 2011

Cf. A034694, A077316, A077317, A077319.

Edited, corrected and extended by Franklin T. Adams-Watters, Aug 29 2006

A093868 Smallest prime that differs from a multiple of n by unity.

Original entry on oeis.org

2, 3, 2, 3, 11, 5, 13, 7, 17, 11, 23, 11, 53, 13, 29, 17, 67, 17, 37, 19, 41, 23, 47, 23, 101, 53, 53, 29, 59, 29, 61, 31, 67, 67, 71, 37, 73, 37, 79, 41, 83, 41, 173, 43, 89, 47, 281, 47, 97, 101, 101, 53, 107, 53, 109, 113, 113, 59, 353, 59, 367, 61, 127, 127, 131, 67, 269

Offset: 1

Amarnath Murthy, Apr 20 2004

Numbers n such that a(n-1)=a(n+1)=n are A025584 (primes p such that p-2 is not a prime). - Rick L. Shepherd, Aug 23 2004

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

Cf. A093869.

Cf. A034694 (Smallest prime == 1 (mod n)), A038700 (Smallest prime == -1 (mod n)).

f:= proc(n) local j,k;
  for k from 1 do
    for j in [-1,1] do
      if isprime(k*n+j) then return k*n+j fi
  od od
end proc:
map(f, [$1..100]); # Robert Israel, Nov 07 2019

a[n_] := Module[{p}, For[p = 2, True, p = NextPrime[p], If[Divisible[p-1, n] || Divisible[p+1, n], Return[p]]]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Feb 04 2023 *)

a(n) = forprime(p=2, , if (!((p+1) % n) || !((p-1) % n), return (p))); \\ Michel Marcus, Aug 08 2014

a(n) = min(A034694(n), A038700(n)) for all n >= 1. - Rick L. Shepherd, Aug 23 2004

More terms from Rick L. Shepherd, Aug 23 2004

cp's OEIS Frontend

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

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A085420 For each n, let p(n,b) be the smallest prime in the arithmetic progression k*n+b, with k > 0. Then a(n) = max(p(n,b)) with 0 < b < n and gcd(b,n) = 1.

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A064632 Smallest prime p such that n = (p-1)/(q-1) for some prime q.

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

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A077316 Triangle read by rows: T(n,k) is the k-th prime = 1 (mod n).

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A034782 Numbers n such that A034693(n) = 3: 3n + 1 is prime, but neither n + 1 nor 2n + 1.

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A034784 Numbers n such that A034693(n) = 2.

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A077317 a(n) is the n-th prime == 1 (mod n).