A057217 -id:A057217 - cp's OEIS frontend

A231119 Least positive k such that n * k^k + 1 is a prime, or 0 if no such k exists.

1, 1, 2, 1, 102414, 1, 2, 17, 2, 1, 36, 1, 2, 3, 2, 1, 210, 1, 20, 3, 990, 1, 6, 2, 2, 6, 2, 1

Offset: 1

Alex Ratushnyak, Nov 04 2013

The number a(5) is conjectured to be zero. Four days of computation have shown that all numbers 5*k^k+1 are composite for k = 1..22733. - T. D. Noe, Nov 11 2013
The sum of 1/log(n*k^k) diverges slowly for every n so normal heuristics predict infinitely many primes in each case, including n=5. - Jens Kruse Andersen, Jun 16 2014
a(5) > 100000 or a(5) = 0. a(29) > 100000 or a(29) = 0. - Jason Yuen, Jan 06 2025
a(5) = 102414 . - Phillip Poplin, May 27 2025
a(29) > 150000 or a(29) = 0. - Phillip Poplin, May 27 2025

Cf. A035093, A057217, A175763, A228175.

import java.math.BigInteger; public class A231119 { public static void main (String[] args) { for (int n = 1; n < 3333; n++) { BigInteger nn = BigInteger.valueOf(n); for (int k=1; k<10000; k++) { BigInteger p = nn.multiply(BigInteger.valueOf(k).pow(k)).add(BigInteger.ONE); if (p.isProbablePrime(80)) { System.out.printf("%d, ", k); break; } else System.out.printf("."); } } } }

a(5) from Phillip Poplin, May 27 2025

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 *)

A251717 a(n) = smallest positive integer k such that A083221(k, n) has at most two prime factors (is a prime or semiprime).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Dec 15 2014

nonn

Records occur at 1, 4, 8, 26, 32, 39, 238, 462, 1075, 1763, ... with record values 1, 2, 3, 4, 5, 6, 8, 9, 11, 13, ...

New distinct values occur at 1, 4, 8, 26, 32, 39, 238, 306, 462, 1075, 1106, 1763, ... with the values 1, 2, 3, 4, 5, 6, 8, 7, 9, 11, 10, 13, ...

Antti Karttunen, Table of n, a(n) for n = 1..2518

Variant: A251718.
The positions of ones: A008578.
Cf. A001222, A083221 (A083140), A251719.
a(n+1) differs from A057217(n-1) for the first time at n=19, where a(20) = 3, while A057217(18) = 4.

(define (A251717 n) (let loop ((i 1)) (if (<= (A001222 (A083221bi i n)) 2) i (loop (+ i 1))))) ;; Code for A083221bi given in A083221.

For all n, a(n) <= A251718(n) <= A251719(n).

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

A231901 Least k > n such that k!/n! + 1 is a prime, or 0 if no such k exists.

Original entry on oeis.org

1, 2, 4, 4, 6, 6, 11, 9, 11, 10, 20, 12, 15, 15, 16, 16, 18, 18, 23, 21, 22, 22, 40, 25, 27, 31, 28, 28, 37, 30, 42, 38, 34, 36, 42, 36, 110, 39, 43, 40, 42, 42, 56, 46, 50, 46, 55, 65, 51, 51, 53, 52, 55, 55, 73, 58, 58, 58, 60, 60, 63, 63, 177, 68, 70, 66, 82, 72

Offset: 0

Alex Ratushnyak, Nov 15 2013

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A002981, A035093, A057217, A075757, A075758, A231549.

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

a(n) = {my(m = n+1); while(! isprime(m!/n! +1), m++); m;} \\ Michel Marcus, Mar 07 2014; corrected Jun 13 2022

A231549 Least k>0 such that k!*n!+1 is a prime, or 0 if no such k exists.

Original entry on oeis.org

1, 1, 1, 4, 2, 8, 3, 3, 3, 4, 1, 2, 3, 5, 8, 4, 10, 2, 11, 9, 5, 5, 7, 3, 14, 18, 1, 40, 24, 5, 5, 18, 8, 20, 2, 49, 1, 3, 5, 28, 1, 17, 38, 27, 11, 16, 10, 3, 24, 270, 2, 45, 2, 15, 175, 64, 17, 6, 4, 3, 8, 18, 13, 17, 65, 32, 12, 7, 72, 13, 21, 33, 1, 24, 36, 76, 1

Offset: 1

Alex Ratushnyak, Nov 15 2013

nonn

Indices of 1's: A002981.

Cf. A002981, A035093, A057217, A075757, A075758, A231901.

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

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 *)

A057218 a(n) = least prime of the form n*k! + 1.

Original entry on oeis.org

2, 3, 7, 5, 11, 7, 43, 17, 19, 11, 23, 13, 79, 29, 31, 17, 103, 19, 457, 41, 43, 23, 47, 577, 151, 53, 163, 29, 59, 31, 156241, 193, 67, 24481, 71, 37, 223, 229, 79, 41, 83, 43, 1033, 89, 271, 47, 283, 97, 5881, 101, 103, 53, 107, 109, 331, 113, 6841, 59, 2355091201, 61

Offset: 1

Labos Elemer, Sep 27 2000

nonn

First prime in the sequence n + 1, 2n + 1, 6n + 1, 24n + 1, 120n + 1, ...

If p is a prime then a(p-1) = p.

			n=275, for k=1,..,24 {1+275*k!}={276,551,.......,170623310476640845824000001} and a(275)=170623310476640845824000001, the prime generated by A057217(275)=24 as 1+275*24!

Primes arising in A057217.

a(n) = k = 1; while (!isprime(p=1+n*k!), k++); p; \\ Michel Marcus, Feb 20 2016

More terms from Mark Hudson (mrmarkhudson(AT)hotmail.com), Aug 12 2004

cp's OEIS Frontend

A231119 Least positive k such that n * k^k + 1 is a prime, or 0 if no such k exists.

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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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A251717 a(n) = smallest positive integer k such that A083221(k, n) has at most two prime factors (is a prime or semiprime).

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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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A231901 Least k > n such that k!/n! + 1 is a prime, or 0 if no such k exists.

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A231549 Least k>0 such that k!*n!+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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A057218 a(n) = least prime of the form n*k! + 1.

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