cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-9 of 9 results.

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

Original entry on oeis.org

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

Views

Author

Alex Ratushnyak, Nov 04 2013

Keywords

Comments

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

Crossrefs

Cf. A035093, A057217, A175763, A228175.

Programs

  • Java
    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("."); } } } }

Extensions

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

Views

Author

Alex Ratushnyak, Nov 13 2013

Keywords

Crossrefs

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).

Programs

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

A083663 Least k such that k*n!-1 is prime.

Original entry on oeis.org

3, 2, 1, 1, 2, 1, 1, 5, 3, 4, 4, 1, 5, 1, 2, 9, 2, 30, 30, 5, 44, 2, 7, 13, 5, 3, 11, 2, 14, 1, 7, 1, 1, 30, 16, 22, 36, 1, 38, 13, 22, 6, 36, 17, 36, 40, 31, 25, 38, 13, 4, 32, 22, 154, 10, 27, 7, 121, 9, 33, 19, 19, 4, 26, 100, 18, 46, 75, 21, 11, 34, 75, 38, 7, 45, 3, 19, 13, 59, 39, 72
Offset: 1

Views

Author

Benoit Cloitre, Jun 14 2003

Keywords

Links

Crossrefs

Cf. A035093, A053989, A084730.

Programs

  • Maple
    f:= proc(n) local w,k;
w:= n!;
for k from 1 do
  if isprime(k*w-1) then return k fi
od
end proc:
map(f, [$1..100]); # Robert Israel, Nov 23 2023
  • Mathematica
    lkp[n_]:=Module[{k=1,nf=n!},While[!PrimeQ[k*nf-1],k++];k]; Array[lkp,100] (* Harvey P. Dale, Apr 26 2025 *)
  • PARI
    a(n)=if(n<1,0,k=1; while(isprime(k*n!-1)==0,k++); k)

Formula

a(n) = A053989(n!) = (A084730(n)+1)/n!. - Robert Israel, Nov 23 2023

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

Views

Author

Alex Ratushnyak, Nov 13 2013

Keywords

Links

Crossrefs

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).

Programs

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

Views

Author

Alex Ratushnyak, Nov 15 2013

Keywords

Links

Crossrefs

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

Programs

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

Views

Author

Alex Ratushnyak, Nov 15 2013

Keywords

Comments

Indices of 1's: A002981.

Crossrefs

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

Programs

  • Mathematica
    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

Views

Author

Alex Ratushnyak, Nov 13 2013

Keywords

Crossrefs

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).

Programs

  • Mathematica
    Table[k = 1; While[! PrimeQ[k*n^n - 1], k++]; k, {n, 65}] (* T. D. Noe, Nov 15 2013 *)

A035094 Smallest prime of form (n!)*k + 1.

Original entry on oeis.org

2, 3, 7, 73, 241, 2161, 15121, 161281, 1088641, 10886401, 39916801, 958003201, 18681062401, 1133317785601, 9153720576001, 83691159552001, 1778437140480001, 12804747411456001, 851515702861824001, 41359334139002880001, 766364132575641600001, 20232013099996938240001
Offset: 1

Views

Author

Labos Elemer

Keywords

Comments

This is one possible generalization of "the least prime problem in special arithmetic progressions" when n in nk+1 is replaced by n!.
a(n) is the smallest prime p such that the multiplicative group modulo p has a subgroup of order n!. - Joerg Arndt, Oct 18 2020

Examples

			a(5)=241 because in arithmetic progression 120k+1=5!k+1 the second term is prime, 241.

Links

Crossrefs

Analogous case is A034694. Special case for k=1 is A002981.
Cf. A035093 (values of k).

Programs

  • Mathematica
    sp[n_]:=Module[{nf=n!,k=1},While[!PrimeQ[nf*k+1],k++];nf*k+1]; Array[sp,20] (* Harvey P. Dale, Jan 27 2013 *)
  • PARI
    a(n) = for(k=1, oo, if(isprime(k*n! + 1), return(k*n! + 1))); \\ Daniel Suteu, Oct 18 2020

A335361 Prime numbers p such that p!*i + 1 is composite for i = 1..p.

Original entry on oeis.org

67, 157, 269, 379, 419, 443, 449, 509, 541, 577, 743, 769, 859, 863, 929, 937, 1009, 1087, 1163, 1213, 1217, 1367, 1381, 1481, 1579, 1733, 1747, 1753, 1783, 1787, 1877, 1901, 1997, 2153
Offset: 1

Views

Author

Chai Wah Wu, Jun 10 2020

Keywords

Comments

Primes in A335233.

Crossrefs

Cf. A035093, A321805, A335233.

Programs

  • Mathematica
    Select[Prime@ Range@ 100, (f = #!; NoneTrue[f*Range[#] + 1, PrimeQ ]) &] (* Robert Price, Sep 14 2020 *)
  • PARI
    is(p) = if(isprime(p), for(i=1, p, if(ispseudoprime(i*p!+1), return(0))); 1, 0); \\ Jinyuan Wang, Jun 21 2020
  • Python
    from sympy import isprime, nextprime, factorial
A335361_list, p = [], 2
while p < 500:
    f, g = factorial(p), 1
    for i in range(1,p+1):
        g += f
        if isprime(g):
            break
    else:
        A335361_list.append(p)
    p = nextprime(p)

Extensions

a(34) from Jinyuan Wang, Jun 21 2020
Showing 1-9 of 9 results.

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