A231119 -id:A231119 - cp's OEIS frontend

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

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 *)

A228175 Least positive k such that n^n * k^k + 1 is a prime, or 0 if no such k exists.

Original entry on oeis.org

1, 1, 1, 2, 1, 6, 5, 2, 7, 10, 8

Offset: 0

Alex Ratushnyak, Nov 02 2013

The next terms after the missing a(11) are 31, 58, 4, 596, 3.
a(11) > 20000 or a(11) = 0, a(17) = 4308, a(18) = 1073, a(19) > 20000 or a(19) = 0. - Jason Yuen, May 21 2024
a(11) > 10^5 or a(11) = 0. a(19) > 10^5 or a(19) = 0. - Jason Yuen, Feb 27 2025

			3^3 * 1 + 1 = 28 is not a prime, 3^3 * 2^2 + 1 = 109 is a prime, so a(3) = 2.

Cf. A000040, A000312, A156223, A231119, A380903.

import java.math.BigInteger;
public class A228175 {
  public static void main (String[] args) {
    for (int n = 0; n < 333; n++) {
      BigInteger nn = BigInteger.valueOf(n).pow(n);
      int k = 1;
      for (; k<10000; k++) {
        BigInteger kk = BigInteger.valueOf(k).pow(k).multiply(nn).add(BigInteger.ONE);
        if (kk.isProbablePrime(80)) {
          System.out.printf("%d, ", k);
          break;
        }
      }
      if (k==10000) System.out.printf("- ");
    }
  }
}

A228175(n,L=9e9,s=1)={forstep(k=s+(bittest(n,0)&&n>1&&bittest(s,0)), L, 1+bittest(n,0), ispseudoprime(n^n*k^k+1)&&return(k))} \\ Optional args allow specification of start and limit for search; for odd n > 1, only check even k. - M. F. Hasler, Nov 03 2013

a(n) = A231119(n^n). - Jason Yuen, Nov 15 2024

A231820 Least positive k such that n*k! - 1 is a prime, or 0 if no such k exists.

Original entry on oeis.org

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

Offset: 1

Alex Ratushnyak, Nov 13 2013

nonn

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

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. A231819 (least k such that k*(n^2)-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. A053989 (least k such that n*k -1    is a prime).

f:= proc(n) local k;
for k from 1 do if isprime(n*k!-1) then return k fi od
end proc:
map(f, [$1..100]); # Robert Israel, Oct 29 2019

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

a(n) = my(k=1); while (!isprime(n*k! - 1), k++); k; \\ Michel Marcus, Oct 29 2019

A231818 Least positive k such that k*n^n - 1 is a prime, or 0 if no such k exists.

Original entry on oeis.org

3, 1, 2, 5, 6, 3, 6, 39, 18, 6, 12, 19, 8, 23, 10, 3, 76, 13, 90, 26, 52, 45, 124, 12, 60, 27, 10, 99, 126, 11, 50, 27, 28, 59, 6, 80, 122, 71, 110, 21, 72, 111, 590, 147, 178, 84, 238, 12, 138, 236, 10, 53, 6, 60, 98, 72, 620, 30, 166, 5, 98, 18, 22, 384, 126

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. A231819 (least k such that k*(n^2)-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^n - 1], k++]; k, {n, 65}] (* T. D. Noe, Nov 15 2013 *)

A380903 Least positive k such that n^n * k^k - 1 is a prime, or 0 if no such k exists.

Original entry on oeis.org

2, 2, 1, 2, 3, 4, 10147, 24

Offset: 0

Jason Yuen, Feb 07 2025

a(8) > 10^5 or a(8) = 0.
a(9) = 0, a(10) = 3, a(11) = 3142, a(12) = 559, a(13) = 3558.
a(14) > 10^5 or a(14) = 0.

			The least k > 0 such that 4^4*k^k - 1 is a prime is k = 3, so a(4) = 3.

Cf. A228175, A231119, A231735.

a(n) = for(k=1, oo, if(ispseudoprime(n^n*k^k-1), return(k))) \\ Does not terminate if a(n) = 0.

a(n) = A231735(n^n).

cp's OEIS Frontend

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

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A228175 Least positive k such that n^n * k^k + 1 is a prime, or 0 if no such k exists.

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A231820 Least positive k such that n*k! - 1 is a prime, or 0 if no such k exists.

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A231818 Least positive k such that k*n^n - 1 is a prime, or 0 if no such k exists.

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Mathematica

A380903 Least positive k such that n^n * k^k - 1 is a prime, or 0 if no such k exists.

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