A034693 -id:A034693 - cp's OEIS frontend

A053989 Smallest k such that nk-1 is prime.

3, 2, 1, 1, 4, 1, 2, 1, 2, 2, 4, 1, 8, 1, 2, 2, 4, 1, 2, 1, 2, 2, 6, 1, 6, 4, 2, 3, 6, 1, 2, 1, 4, 2, 4, 2, 2, 1, 6, 2, 4, 1, 6, 1, 2, 3, 6, 1, 2, 3, 2, 2, 4, 1, 2, 3, 2, 3, 6, 1, 8, 1, 4, 2, 6, 2, 6, 1, 2, 2, 4, 1, 14, 1, 2, 2, 4, 3, 2, 1, 8, 2, 4, 1, 6, 3, 2, 3, 16, 1, 2, 4, 6, 3, 4, 2, 2, 1, 2, 2

Offset: 1

Henry Bottomley, Apr 04 2000

			a(5)=4 because the smallest prime in the sequence 5k-1 (4,9,14,19,24...) is 19 when k=4

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A034693, A038700, A071558, A010051, A103689, A200996.

a053989 n = head [k | k <- [1..], a010051' (k * n - 1) == 1]
-- Reinhard Zumkeller, Feb 14 2013

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

a(n) = (A038700(n)+1)/n.

A071558 Smallest k such that nk + 1 and nk - 1 are twin primes.

Original entry on oeis.org

4, 2, 2, 1, 6, 1, 6, 9, 2, 3, 18, 1, 24, 3, 2, 12, 6, 1, 12, 3, 2, 9, 6, 3, 6, 12, 4, 15, 12, 1, 42, 6, 6, 3, 12, 2, 54, 6, 8, 6, 30, 1, 24, 15, 4, 3, 6, 4, 18, 3, 2, 6, 120, 2, 12, 48, 4, 6, 18, 1, 258, 21, 14, 3, 30, 3, 24, 15, 2, 6, 18, 1, 84, 27, 2, 3, 6, 4, 132, 3, 10, 15, 54, 5, 12, 12

Offset: 1

Benoit Cloitre, May 30 2002

Conjecture: a(n) < sqrt(n)*log(n) for all n > 17261. This has been verified for n up to 3*10^7. It implies the inequality a(n) < n for each n > 127. - Zhi-Wei Sun, Jan 07 2013

A200996(n) <= a(n). - Reinhard Zumkeller, Feb 14 2013

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

Cf. A071256, A001097, A086686, A034693.
Cf. A071407 (k at prime n).
Cf. A220143, A220144 (record values).

a071558 n = head [k | k <- [1..], let x = k * n,
                      a010051' (x - 1) == 1, a010051' (x + 1) == 1]
-- Reinhard Zumkeller, Feb 14 2013

Table[k=1; While[!And@@PrimeQ[n*k+{1,-1}],k++]; k,{n,86}] (* Jayanta Basu, May 26 2013 *)

a(n) = my(s=1); while(isprime(s*n+1)*isprime(n*s-1)==0, s++); s;

A035096 a(n) is the smallest k such that prime(n)*k+1 is prime.

Original entry on oeis.org

1, 2, 2, 4, 2, 4, 6, 10, 2, 2, 10, 4, 2, 4, 6, 2, 12, 6, 4, 8, 4, 4, 2, 2, 4, 6, 6, 6, 10, 2, 4, 2, 6, 4, 8, 6, 10, 4, 14, 2, 2, 6, 2, 4, 18, 4, 10, 12, 24, 12, 2, 2, 6, 2, 6, 6, 8, 6, 4, 2, 6, 2, 4, 6, 6, 26, 6, 10, 6, 10, 14, 2, 6, 4, 12, 12, 24, 6, 8, 4, 2, 10, 2, 4, 10, 2, 8, 30

Offset: 1

Labos Elemer

nonn

These arithmetic progressions have prime differences. Note that both the terms of generated by this k values and the differences are primes as well.
This is one possible generalization of "the least prime problem in special arithmetic progressions" when n in the nk+1 form is replaced by n-th prime number.
Note that Dirichlet's theorem on primes in arithmetic progressions implies that a(n) always exists. - Max Alekseyev, Jul 11 2008
If a(n)=2, prime(n) is a Sophie Germain prime (A005384). Among the first 10^6 terms, the largest is a(330408) = 234. - Zak Seidov, Jan 28 2012

			a(15)=6 because the 15th prime is 47, and the smallest k such that 47k+1 is prime is k=6, for which 47k+1=283.

T. D. Noe, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Math, Dirichlet's theorem.
Index entries for sequences related to primes in arithmetic progressions.

Smallest k such that k*n+1 is prime is A034693.
Sophie Germain primes are in A005384.
Cf. A000040, A035095. - Zak Seidov, Dec 27 2013
Cf. A117673.

S:=[];
k:=1;
for n in [1..90] do
  while not IsPrime(k*NthPrime(n)+1) do
       k:=k+1;
  end while;
  Append(~S, k);
  k:=1;
end for;
S; // Bruno Berselli, Apr 18 2013

Reap[Sow[1]; Do[p = Prime[n]; k = 2; While[! PrimeQ[k*p + 1], k = k + 2]; Sow[k], {n, 2, 10^4}]][[2, 1]] (* Zak Seidov, Jan 28 2012 *)
f[n_] := Block[{p = Prime@ n}, q = 1 + 2p; While[ !PrimeQ@ q, q += 2p]; (q - 1)/p]; f[1] = 1; Array[f, 88] (* Robert G. Wilson v, Dec 27 2014 *)

a(n) = if(n == 1, 1, my(t = 2*prime(n), m = t + 1); while(!isprime(m), m += t); 2*(m - 1)/t); \\ Amiram Eldar, Mar 19 2025

a(n) = (A035095(n)-1)/A000040(n). - Zak Seidov, Dec 27 2013

A035092 Smallest k such that (n^2)*k + 1 is prime.

Original entry on oeis.org

1, 1, 2, 1, 4, 1, 4, 3, 2, 1, 6, 3, 4, 1, 8, 1, 12, 4, 30, 1, 2, 3, 24, 1, 18, 1, 2, 4, 12, 2, 16, 12, 2, 3, 6, 1, 4, 13, 6, 1, 10, 2, 12, 6, 2, 6, 4, 8, 6, 9, 6, 9, 28, 1, 4, 1, 10, 3, 6, 4, 46, 4, 4, 3, 4, 1, 4, 3, 22, 6, 10, 2, 4, 1, 2, 7, 22, 3, 6, 4, 6, 3, 10, 1, 4, 3, 2, 4, 6, 1, 10, 4, 2, 1

Offset: 1

Labos Elemer

			a(40) = 1 because in 1600k + 1 at k = 1, 1601 is the smallest prime;
a(61) = 46 because in the 46*46*k + 1 sequence the first prime appears at k = 46; it is 171167.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Index entries for sequences related to primes in arithmetic progressions

Analogous case is A034693. See also A005574 and A002496.

Table[k = 1; While[! PrimeQ[k (n^2) + 1], k++]; k, {n, 94}] (* Michael De Vlieger, Dec 17 2016 *)

a(n)=k=1;while(!isprime(k*n^2+1),k++);k
vector(100,n,a(n)) \\ Derek Orr, Oct 01 2014

A035093 Smallest k such that k*n! + 1 is prime.

Original entry on oeis.org

1, 1, 1, 3, 2, 3, 3, 4, 3, 3, 1, 2, 3, 13, 7, 4, 5, 2, 7, 17, 15, 18, 3, 6, 3, 16, 1, 4, 7, 20, 8, 3, 9, 5, 2, 7, 1, 3, 10, 3, 1, 29, 7, 9, 45, 8, 3, 6, 35, 66, 2, 20, 2, 4, 25, 52, 14, 34, 24, 6, 10, 22, 38, 16, 20, 91, 69, 12, 19, 20, 21, 42, 1, 5, 33, 77, 1, 2, 12, 29, 193, 74, 40, 55, 19

Offset: 1

Labos Elemer

nonn

This is one possible generalization of "the least prime problem" for n*k+1 arithmetic progression when n is replaced by n!, a special difference.

			a(7)=3 because in progression of 5040*k+1 the terms 5041 and 10081 are not prime and so 15121 is the first prime.

Index entries for sequences related to primes in arithmetic progressions

Analogous case is A034693. Special case for k=1 is A002981.

Table[k = 1; While[! PrimeQ[1 + k*n!], k++]; k, {n, 85}] (* T. D. Noe, Nov 04 2013 *)

a(n) = my(k=1); while(!isprime(k*n!+1), k++); k; \\ Michel Marcus, Sep 26 2020

a(80) corrected by Alex Ratushnyak, Nov 03 2013

Simpler title by Sean A. Irvine, Sep 25 2020

A103689 a(n) is the least k such that either kn - 1 or kn + 1 (or both) is prime.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 4, 1, 2, 1, 2, 1, 2, 1, 4, 2, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 6, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 6, 1, 6, 1, 2, 2, 2, 1, 4, 1, 2, 1, 4, 1, 4, 1, 2, 2, 4, 1, 2, 1, 2, 1, 2, 1, 6, 2, 2, 1, 2, 1, 2, 3, 4, 3, 2, 1, 2, 1, 2, 1, 6, 1, 6, 1, 2

Offset: 1

Pierre CAMI, Feb 12 2005

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A002110, A034693, A053989, A103688, A200996, A348347.

a103689 n = min (a053989 n) (a034693 n)
-- Reinhard Zumkeller, Feb 14 2013

f[n_] := Block[{k = 1}, While[ ! PrimeQ[k*n - 1] && ! PrimeQ[k*n + 1], k++ ]; k]; Table[ f[n], {n, 105}] (* Robert G. Wilson v, Feb 12 2005 *)
lk[n_]:=Module[{k=1},While[NoneTrue[k*n+{1,-1},PrimeQ],k++];k]; Array[ lk,120] (* The program uses the NoneTrue function from Mathematica version 10 *) (* Harvey P. Dale, May 01 2016 *)

a(n) = my(k=1); while (!isprime(k*n+1) && !isprime(k*n-1), k++); k; \\ Michel Marcus, Oct 18 2021

a(n) <= A200996(n). - Reinhard Zumkeller, Feb 14 2013
a(n) = min {A053989(n), A034693(n)}. - Reinhard Zumkeller, Feb 14 2013
a(A002110(n)/3+3) >= ceiling((prime(n+1)-1)/3) for n >= 2. Equality holds for n = 2, 4, 6, 8, 10, 12, 22, 25, 31, 116, 155, 156, 197, ... . - Pontus von Brömssen, Oct 16 2021
a(A002110(n)/3-3) >= ceiling((prime(n+1)-1)/3) for n >= 3. Equality holds for n = 3, 4, 5, 6, 7, 9, 39, 51, 59, 65, 98, 311, ... . - Pontus von Brömssen, Oct 19 2021

Edited, corrected and extended by Robert G. Wilson v, Feb 19 2005

A231819 Least positive k such that k*n^2 - 1 is a prime, or 0 if no such k exists.

Original entry on oeis.org

3, 1, 2, 2, 6, 2, 2, 2, 8, 2, 2, 3, 2, 3, 2, 5, 2, 2, 8, 5, 2, 2, 8, 2, 2, 3, 6, 2, 12, 3, 8, 5, 10, 2, 6, 2, 12, 2, 2, 3, 2, 2, 2, 3, 2, 2, 18, 3, 2, 2, 8, 2, 20, 3, 6, 2, 18, 3, 2, 3, 12, 2, 2, 2, 6, 7, 8, 6, 2, 3, 14, 3, 2, 3, 6, 2, 6, 3, 8, 2, 2, 5, 6, 5, 2

Offset: 1

Alex Ratushnyak, Nov 13 2013

nonn

Cf. A035092 (least k such that k*(n^2)+1 is a prime).
Cf. A175763 (least k such that k*(n^n)+1 is a prime).
Cf. A035093 (least k such that k*n!+1    is a prime).
Cf. A193807 (least k such that n*(k^2)+1 is a prime).
Cf. A231119 (least k such that n*(k^k)+1 is a prime).
Cf. A057217 (least k such that n*k!+1    is a prime).
Cf. A034693 (least k such that n*k +1    is a prime).
Cf. A231818 (least k such that k*(n^n)-1 is a prime).
Cf. A083663 (least k such that k*n!-1    is a prime).
Cf. A231734 (least k such that n*(k^2)-1 is a prime).
Cf. A231735 (least k such that n*(k^k)-1 is a prime).
Cf. A231820 (least k such that n*k!-1    is a prime).
Cf. A053989 (least k such that n*k -1    is a prime).

Table[k = 1; While[! PrimeQ[k*n^2 - 1], k++]; k, {n, 100}] (* T. D. Noe, Nov 18 2013 *)

A200996 Least upper limit for numbers u and v such that un-1 and vn+1 are both prime, u and v not necessarily distinct.

Original entry on oeis.org

3, 2, 2, 1, 4, 1, 4, 2, 2, 2, 4, 1, 8, 2, 2, 2, 6, 1, 10, 2, 2, 2, 6, 3, 6, 4, 4, 3, 6, 1, 10, 3, 4, 3, 4, 2, 4, 5, 6, 2, 4, 1, 6, 2, 4, 3, 6, 2, 4, 3, 2, 2, 4, 2, 6, 3, 4, 3, 12, 1, 8, 5, 4, 3, 6, 2, 6, 2, 2, 2, 8, 1, 14, 2, 2, 3, 6, 3, 4, 3, 8, 2, 4, 4, 12

Offset: 1

Reinhard Zumkeller, Feb 14 2013

nonn

A103689(n) <= a(n) <= A071558(n).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Haskell
```
a200996 n = max (a053989 n) (a034693 n)
```

a(n) = max {A053989(n), A034693(n)}.

A363533 Least k such that n*F(k)+1 is prime, where F = A000045 is the Fibonacci sequence, or -1 if no such k exists.

Original entry on oeis.org

1, 1, 3, 1, 3, 1, 9, 3, 3, 1, 3, 1, 9, 3, 3, 1, 6, 1, 9, 3, 3, 1, 3, 4, 18, 3, 9, 1, 3, 1, 15, 4, 3, 4, 3, 1, 9, 5, 3, 1, 3, 1, 48, 3, 9, 1, 24, 3, 9, 3, 3, 1, 3, 3, 9, 3, 6, 1, 24, 1, 36, 5, 3, 4, 3, 1, 12, 3, 3, 1, 6, 1, 12, 3, 3, 4, 6, 1, 9, 4, 3, 1, 3, 5

Offset: 1

Pontus von Brömssen, Jun 09 2023

nonn

2 does not appear because F(1) = F(2).

a(n) is divisible by 3 if n >= 3 is odd (unless a(n) = -1), because F(k) is odd (so n*F(k)+1 > 2 is even) when k is not divisible by 3.

			For n = 17, the least k such that 17*F(k)+1 is prime is k = 6, with 17*F(6)+1 = 17*8+1 = 137, so a(17) = 6.

Cf. A000045, A034693, A124067, A361902, A362376, A363534 (records), A363535 (indices of records), A363536 (first occurrences).

Array[(k = 1; While[! PrimeQ[# Fibonacci[k] + 1], k++]; k) &, 85] (* Michael De Vlieger, Jun 10 2023 *)

a(n) = my(k=1); while(!isprime(n*fibonacci(k)+1), k++); k; \\ Michel Marcus, Jun 10 2023

from sympy import isprime, fibonacci
from itertools import count
def A363533(n):
    # Note: the function hangs if a(n) = -1.
    return next(k for k in count(1) if isprime(n*fibonacci(k)+1))

a(n) = 1 if and only if n+1 is prime.

A034849 a(n) = 1 + 2*A034784(n).

Original entry on oeis.org

7, 11, 17, 19, 23, 29, 31, 41, 43, 47, 53, 59, 67, 71, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 163, 167, 173, 179, 181, 191, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 281, 283, 293, 307, 311, 317, 331

Offset: 1

Labos Elemer

nonn

The terms are odd prime numbers.

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

Cf. A034693, A034694, A034784.

1 + 2 Position[#, 2] &@ Table[k = 1; While[! PrimeQ[k n + 1], k++]; k, {n, 165}] // Flatten (* Michael De Vlieger, Jul 31 2017 *)

lista(nn) = for (n=1, nn, if (isprime(p=2*n+1) && !isprime(n+1), print1(p, ", "));); \\ Michel Marcus, Aug 01 2017

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

cp's OEIS Frontend

A053989 Smallest k such that nk-1 is prime.

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A071558 Smallest k such that n*k + 1 and n*k - 1 are twin primes.

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A035096 a(n) is the smallest k such that prime(n)*k+1 is prime.

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A035092 Smallest k such that (n^2)*k + 1 is prime.

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A035093 Smallest k such that k*n! + 1 is prime.

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A103689 a(n) is the least k such that either k*n - 1 or k*n + 1 (or both) is prime.

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A231819 Least positive k such that k*n^2 - 1 is a prime, or 0 if no such k exists.

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A200996 Least upper limit for numbers u and v such that u*n-1 and v*n+1 are both prime, u and v not necessarily distinct.

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A071558 Smallest k such that nk + 1 and nk - 1 are twin primes.

A103689 a(n) is the least k such that either kn - 1 or kn + 1 (or both) is prime.

A200996 Least upper limit for numbers u and v such that un-1 and vn+1 are both prime, u and v not necessarily distinct.