A231819 -id:A231819 - cp's OEIS frontend

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

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

A110559 Least j such that jn^2 -1 and jn^2 +1 are twin primes.

Original entry on oeis.org

4, 1, 2, 12, 6, 2, 18, 3, 10, 6, 12, 3, 12, 18, 18, 57, 12, 5, 120, 12, 2, 3, 132, 2, 42, 3, 58, 45, 12, 7, 72, 15, 10, 3, 6, 2, 60, 30, 12, 3, 168, 2, 192, 18, 2, 33, 48, 10, 138, 39, 8, 63, 42, 22, 60, 42, 32, 3, 120, 6, 90, 18, 40, 165, 204, 7, 90, 18, 70, 6, 72, 27, 30, 15, 6, 18

Offset: 1

Pierre CAMI, Sep 12 2005

nonn

Define Sj=sum of j(n) for n=1 to N. Define Sn=sum of (2*log(n))^2 for n=1 to N. As N increases Sj/Sn tends to 0.6. - Pierre CAMI, Dec 13 2011

			12*4*4-1=191, 191 and 193 are twin primes so a(4)=12.

Pierre CAMI, Table of n, a(n) for n = 1..10000

Cf. A231819.

a(n) = my(j=1); while (!(isprime(p=j*n^2-1) && isprime(p+2)), j++); j; \\ Michel Marcus, Sep 17 2019

Extended by Ray Chandler, Sep 15 2005

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 *)

A238848 Smallest k such that k*n^3 - 1 is prime.

Original entry on oeis.org

3, 1, 2, 2, 4, 2, 14, 7, 6, 2, 4, 4, 14, 3, 4, 2, 16, 4, 12, 9, 2, 5, 16, 2, 2, 3, 16, 6, 10, 4, 2, 4, 22, 2, 6, 3, 6, 10, 6, 3, 22, 5, 2, 3, 4, 2, 18, 4, 26, 10, 4, 5, 6, 2, 2, 7, 6, 2, 10, 5, 2, 9, 4, 2, 16, 3, 6, 9, 2, 3, 30, 5, 14, 6, 24, 5, 16, 5

Offset: 1

Derek Orr, Mar 06 2014

nonn

			a(1) = 3 because for k = 1, 1*(1^3) - 1 = 0 is not prime, for k = 2, 2*(1^3) - 1 = 1 is not prime, but for k = 3, 3*(1^3) - 1 = 2 is prime.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A231819, A053989.

sk[n_]:=Module[{k=1,n3=n^3},While[!PrimeQ[k*n3-1],k++];k]; Array[sk,80] (* Harvey P. Dale, Jan 04 2023 *)

import sympy
from sympy import isprime
def f(n):
  for k in range(1,10**3):
    if isprime(k*(n**3)-1):
      return k
n = 1
while n < 10**3:
  print(f(n))
  n += 1

A239020 Smallest number k such that kn +/- 1 and kn^2 +/- 1 are two sets of twin primes. a(n) = 0 if no such number exists.

Original entry on oeis.org

4, 3, 2, 15, 6, 2, 150, 75, 20, 6, 78, 85, 2490, 30, 18, 195, 5160, 490, 330, 12, 2, 870, 330, 13, 42, 105, 2280, 375, 12, 41, 1632, 720, 90, 3, 216, 2, 1380, 615, 98, 84, 438, 65, 600, 210, 148, 735, 3870, 115, 138, 39, 182, 2715, 16590, 48, 60, 63, 210, 120

Offset: 1

Derek Orr, Mar 09 2014

nonn

If n>3 is odd and not a multiple of 3, then a(n) is a multiple of 6; e.g., a(5) = 6, a(7) = 150, a(11) = 78. If n>3 is even and not a multiple of 3, then a(n) is a multiple of 3. In short, for n>1, k*n should be a multiple of 6. - Zak Seidov, Mar 13 2014

			1*2 +/- 1 (1 and 3) and 1*2^2 +/- 1 (3 and 5) are not two sets of twin primes. 2*2 +/- 1 (3 and 5) and 2*2^2 +/- 1 (7 and 9) are not two sets of twin primes. However, 3*2 +/- 1 (5 and 7) and 3*2^2 +/- 1 (11 and 13) are two sets of twin primes. Thus, a(2) = 3.

Cf. A231819, A053989, A035092, A034693.

a(n) = {k = 1; while (! (isprime(k*n+1) && isprime(k*n-1) && isprime(k*n^2+1) && isprime(k*n^2-1)), k++); k;} \\ Michel Marcus, Mar 15 2014

from sympy import isprime
def b(n):
  for k in range(10**5):
    if isprime(k*n+1) and isprime(k*n-1) and isprime(k*(n**2)+1) and isprime(k*(n**2)-1):
      return k
n = 1
while n < 100:
  print(b(n))
  n += 1

A239021 Smallest number k such that kn +/- 1, kn^2 +/- 1, and k*n^3 +/- 1 are three sets of twin primes. a(n) = 0 if no such number exists.

Original entry on oeis.org

4, 105525, 10990, 15855, 344190, 2, 74580, 11580, 165592, 3759, 204918, 12670, 99090, 78, 3978, 11655, 8979180, 10605, 55188, 1221, 2, 23340, 4431420, 39158, 58464, 87318, 45420, 15780, 210, 91, 289422, 19740, 186410, 1293, 137664, 747, 443730, 94920, 278278

Offset: 1

Derek Orr, Mar 09 2014

nonn

			1*6 +/- 1 (5 and 7), 1*6^2 +/- 1 (35 and 37), and 1*6^3 +/- 1 (215 and 217) are not three sets of twin primes. However, 2*6 +/- 1 (11 and 13), 2*6^2 +/- 1 (71 and 73), and 2*6^3 +/- 1 (431 and 433) are three sets of twin primes. Thus, a(6) = 2.

Cf. A238847, A238848, A231819, A053989, A035092, A034693.

from sympy import isprime
def b(n):
  for k in range(10**8):
    if isprime(k*n+1) and isprime(k*n-1) and isprime(k*(n**2)+1) and isprime(k*(n**2)-1) and isprime(k*(n**3)+1) and isprime(k*(n**3)-1):
      return k
n = 1
while n < 100:
  print(b(n))
  n += 1

cp's OEIS Frontend

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

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PARI

A110559 Least j such that j*n^2 -1 and j*n^2 +1 are twin primes.

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PARI

Extensions

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

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Programs

Mathematica

A238848 Smallest k such that k*n^3 - 1 is prime.

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Mathematica

Python

A239020 Smallest number k such that k*n +/- 1 and k*n^2 +/- 1 are two sets of twin primes. a(n) = 0 if no such number exists.

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Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Python

A239021 Smallest number k such that k*n +/- 1, k*n^2 +/- 1, and k*n^3 +/- 1 are three sets of twin primes. a(n) = 0 if no such number exists.

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Author

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Examples

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Programs

Python

A110559 Least j such that jn^2 -1 and jn^2 +1 are twin primes.

A239020 Smallest number k such that kn +/- 1 and kn^2 +/- 1 are two sets of twin primes. a(n) = 0 if no such number exists.

A239021 Smallest number k such that kn +/- 1, kn^2 +/- 1, and k*n^3 +/- 1 are three sets of twin primes. a(n) = 0 if no such number exists.