A116954 -id:A116954 - cp's OEIS frontend

A239666 a(n) = least number k such that n*k^n+1 is prime, or 0 if no such number exists.

1, 1, 4, 1, 8, 1, 4, 3, 10, 1, 42, 1, 60, 15, 22, 1, 8, 1, 198, 42, 10, 1, 8, 115, 34, 21, 0, 1, 54, 1, 130, 3, 4, 7, 72, 1, 778, 204, 30, 1, 108, 1, 178, 15, 14, 1, 924, 28, 234, 63, 1376, 1, 44, 3, 16, 27, 256, 1, 180, 1, 706, 51, 98, 0, 546, 1, 4, 153, 150, 1, 170

Offset: 1

Derek Orr, Mar 29 2014

nonn

a(n) = 1 iff n+1 is prime.
If a(n) = 0, then n is in A097792. Note that the converse is not true: a(4) = 1, not 0.
If n is in A097792 and n > 4, then a(n) = 0. For a sketch of this proof, either n = 4b^4 for some positive integer b > 2 or n = (bp)^p for some prime p > 2 and some positive integer b. In the first case, n*k^n+1 can be factored by Sophie Germain's identity into two trinomials where neither can equal 1 since b > 2, so n*k^n+1 must be composite. In the second case, (bpk^{b^p p^(p-1)}+1) is a factor of n*k^n+1 since p is odd. - William Dean, Oct 23 2024

			3*1^3+1 = 4 is not prime. 3*2^3+1 = 25 is not prime. 3*3^3+1 = 82 is not prime. 3*4^3+1 = 193 is prime. Thus, a(3) = 4.

William Dean, Table of n, a(n) for n = 1..1000

Cf. A097792, A116954, A089001.

is_A097792(n)={my(p,t); n%4==0 && ispower(n\4, 4) || ((2 < p = ispower(n, , &t)) && if(isprime(p), t%p==0, foreach(factor(p)[, 1], q, q%2 && n%q==0 && return(1))))}
a(n) = if(n!=4 && is_A097792(n), 0, for(k=1,oo,if(ispseudoprime(n*k^n+1),return(k)))); \\ [corrected by Andrew Howroyd, Oct 25 2024]

A239787 Numbers n such that 3n^3 - 1 is prime.

Original entry on oeis.org

1, 2, 4, 6, 10, 14, 26, 34, 36, 40, 44, 54, 64, 66, 70, 84, 86, 92, 100, 102, 116, 122, 134, 146, 150, 152, 156, 160, 172, 174, 176, 180, 190, 194, 200, 216, 252, 254, 262, 274, 306, 320, 322, 334, 336, 350, 352, 374, 384, 402, 414, 440, 442, 444, 460

Offset: 1

Derek Orr, Mar 26 2014

All the numbers in this sequence, excluding a(1), are even.

			3*2^3 - 1 = 23 is prime. Thus, 2 is a member of this sequence.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A116954.

[n: n in [0..1000] | IsPrime(3*n^3-1)]; // Vincenzo Librandi, Mar 29 2014

Select[Range[1000], PrimeQ[3 #^3 - 1] &] (* Vincenzo Librandi, Mar 29 2014 *)

s=[]; for(n=1, 1000, if(isprime(3*n^3-1), s=concat(s, n))); s \\ Colin Barker, Mar 27 2014

import sympy
from sympy import isprime
{print(n) for n in range(10**3) if isprime(3*(n**3)-1)}

A240234 Least number k such that k*n^k + 1 is prime, or 0 if no such number exists.

Original entry on oeis.org

1, 1, 2, 1, 1242, 1, 34, 5, 2, 1, 10, 1

Offset: 1

Derek Orr, Apr 02 2014

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

Next term a(13), if it exists, is greater than 1000000. Other unknown terms are at index: 25, 29, 41, 47, 49, 53, 55, 69, 73, 79. (See links). - Jeppe Stig Nielsen, Aug 08 2014

			1*3^1 + 1 = 4 is not prime. 2*3^2 + 1 = 19 is prime. Thus, a(3) = 2.

Guenter Loeh, Generalized Cullen primes n b^n + 1
Jeppe Stig Nielsen, Table of a(n) for n = 1..100, with unknown entries denoted by question marks

Cf. A089001, A116954.

a(n) = for(k=1,4000,if(ispseudoprime(k*n^k+1),return(k)))
n=1;while(n<100,print1(a(n),", ");n++)

A239788 Numbers n such that 3n^3 +/- 1 are twin primes.

Original entry on oeis.org

4, 10, 14, 36, 54, 64, 70, 86, 150, 174, 176, 180, 200, 306, 384, 440, 494, 650, 706, 800, 824, 924, 976, 980, 986, 1020, 1026, 1054, 1360, 1464, 1504, 1506, 1536, 1564, 1604, 1680, 1724, 1736, 2066, 2076, 2116, 2134, 2136, 2166, 2200, 2220, 2314, 2380, 2456

Offset: 1

Derek Orr, Mar 26 2014

Numbers in this sequence are all even.

			3*4^3-1 = 191 is prime and 3*4^3+1 = 193 is prime. Thus, 4 is a member of this sequence.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A116954, A239787.

[n: n in [0..5000] | IsPrime(3*n^3-1) and IsPrime(3*n^3+1)]; // Vincenzo Librandi, Mar 29 2014

Select[Range[5000], PrimeQ[3 #^3 - 1] && PrimeQ[3 #^3 + 1]&] (* Vincenzo Librandi, Mar 29 2014 *)

s=[]; for(n=1, 3000, if(isprime(3*n^3-1) && isprime(3*n^3+1), s=concat(s, n))); s \\ Colin Barker, Mar 27 2014

import sympy
from sympy import isprime
{print(n) for n in range(10**4) if isprime(3*(n**3)+1) and isprime(3*(n**3)-1)}

cp's OEIS Frontend

A239666 a(n) = least number k such that n*k^n+1 is prime, or 0 if no such number exists.

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A239787 Numbers n such that 3n^3 - 1 is prime.

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Magma

Mathematica

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A240234 Least number k such that k*n^k + 1 is prime, or 0 if no such number exists.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A239788 Numbers n such that 3n^3 +/- 1 are twin primes.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Python