A028879 -id:A028879 - cp's OEIS frontend

A137270 Primes p such that p^2 - 6 is also prime.

3, 5, 7, 13, 17, 23, 47, 53, 67, 73, 83, 97, 107, 113, 167, 193, 197, 263, 293, 317, 367, 373, 383, 457, 463, 467, 487, 503, 557, 593, 607, 643, 647, 673, 677, 683, 773, 787, 797, 823, 827, 857, 877, 887, 947, 1033, 1063, 1087, 1103, 1187, 1193, 1223, 1303

Offset: 1

Ben de la Rosa and Johan Meyer (meyerjh.sci(AT)ufa.ac.za), Mar 13 2008

Each of the primes p = 2,3,5,7,13 has the property that the quadratic polynomial phi(x) = x^2 + x - p^2 takes on only prime values for x = 1,2,...,2p-2; each case giving exactly one repetition, in phi(p-1) = -p and phi(p) = p.

The only common term in A062718 and A137270 is 5. - Zak Seidov, Jun 16 2015

			The (2 x 7 - 2) -1 = 11 primes given by the polynomial x^2 + x - 7^2 for x = 1, 2, ..., 2 x 7 - 2 are -47, -43, -37, -29, -19, -7, 7, 23, 41, 61, 83, 107.

F. G. Frobenius, Uber quadratische Formen, die viele Primzahlen darstellen, Sitzungsber. d. Konigl. Acad. d. Wiss. zu Berlin, 1912, 966 - 980.

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

Cf. A062326, A062718.

[p: p in PrimesUpTo(1350) | IsPrime(p^2-6)]; // Vincenzo Librandi, Apr 14 2013

isA028879 := proc(n) isprime(n^2-6) ; end: isA137270 := proc(n) isprime(n) and isA028879(n) ; end: for i from 1 to 300 do if isA137270(ithprime(i)) then printf("%d, ",ithprime(i)) ; fi ; od: # R. J. Mathar, Mar 16 2008

Select[Prime[Range[2,300]],PrimeQ[#^2-6]&] (* Harvey P. Dale, Jul 24 2012 *)

A000040 INTERSECT A028879. - R. J. Mathar, Mar 16 2008

Corrected and extended by R. J. Mathar, Mar 16 2008

A028880 Primes of the form n^2 - 6.

Original entry on oeis.org

3, 19, 43, 163, 283, 523, 619, 2203, 2803, 3019, 4219, 4483, 5323, 5923, 6883, 7219, 9403, 11443, 12763, 13219, 15619, 17683, 20443, 21019, 24019, 27883, 34963, 37243, 38803, 41203, 42019, 46219, 55219, 69163, 75619, 85843, 100483

Offset: 1

Patrick De Geest

a(n) == 19 mod 24 for all n > 1. - Zak Seidov, Mar 10 2015

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
P. De Geest, Palindromic Quasipronics of the form n(n+x)

[ a: n in [1..250] | IsPrime(a) where a is 4*n^2+4*n-5 ]; // Vincenzo Librandi, Aug 05 2010

for(n=1,500,if(isprime(p=n^2-6),print1(p,", "))) \\ Derek Orr, Mar 11 2014

a(n) = A028879(n)^2 - 6. - Zak Seidov, Mar 10 2015

A382246 Smallest number k such that k^n - 6 is prime.

Original entry on oeis.org

8, 3, 2, 5, 5, 5, 19, 85, 7, 5, 19, 275, 23, 43, 53, 455, 65, 23, 23, 175, 7, 65, 47, 295, 7, 143, 49, 115, 23, 355, 185, 305, 7, 55, 319, 85, 113, 25, 329, 505, 25, 187, 205, 25, 295, 437, 17, 2285, 7, 583, 35, 1375, 5, 7, 35, 895, 235, 277, 197, 695, 203, 145, 43, 35, 437, 215

Offset: 1

Jakub Buczak, Mar 19 2025

nonn

No term k in the sequence can be divisible by 2 or 3. Except for the special case a(1)-a(3), where the result of k^n - 6 is either the prime number 2 or 3.
If n is a multiple of 4, the only valid terms of k are those ending in a 5.
Empirical analysis suggests that the terms are typically prime or semiprime.

			a(1) = 8, because 8^1 - 6 = 2, which is prime.
a(4) = 5, because 5^4 - 6 = 619, which is prime.

Jakub Buczak, Table of n, a(n) for n = 1..500

Cf. A380905

Cf. A028879 (a(2)), A239414 (a(6)) for the first term.

a(n) = my(k=1); while (!isprime(k^n-6), k++); k; \\ Michel Marcus, Mar 19 2025

from sympy import isprime
def a(n):
    k = 1
    while (n>1 and k not in [2,3] and (k%2==0 or k%3==0)) or not isprime(k**n-6):
        k += 1
    return k

A309726 Numbers k such that k^2 - 12 is prime.

Original entry on oeis.org

5, 7, 11, 13, 17, 19, 25, 29, 35, 41, 49, 53, 59, 61, 79, 85, 91, 95, 97, 103, 107, 113, 119, 121, 137, 139, 145, 149, 163, 169, 173, 179, 181, 185, 191, 205, 209, 227, 233, 235, 245, 251

Offset: 1

Daniel Starodubtsev, Aug 14 2019

nonn

All terms are odd and not divisible by 3.

			11 is in the sequence because 11^2 - 12 = 109, which is prime.

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

Cf. A028870, A028873, A028876, A028879, A028882, A028885, A186815, A090696, A296507.

Cf. A056927.

Select[Range[5,301,2],PrimeQ[#^2-12]&] (* Harvey P. Dale, Dec 23 2019 *)

select(n->isprime(n^2-12), [1..1000]) \\ Andrew Howroyd, Aug 14 2019

If A056927(k) = 12, then k is a term. - A.H.M. Smeets, Aug 15 2019

A380905 Smallest number k such that k^(2*3^n) - 6 is prime.

Original entry on oeis.org

3, 5, 23, 7, 433, 2447, 9377, 82597, 134687

Offset: 0

Jakub Buczak, Feb 07 2025

Terms must have an ending digit of 3, 5 or 7. If k ends in 1 or 9, then k^(2*3^n)-6 ends in a 5, which is not prime.

a(7) is the first composite term. - Michael S. Branicky, Feb 24 2025

			For n=0, k^(2*3^0) - 6 is prime for the first time at a(0) = k = 3.
For n=5, k^(2*3^5) - 6 is prime for the first time at a(5) = k = 2447.

Cf. Subsequence of A382246.
Cf. A008776, A025192.
Cf. A028879 (a(0)), A239414 (a(1)) for the first term.

a(n) = my(p=3,q=2*3^n); while (!ispseudoprime(p^q-6), p+=2); p; \\ Michel Marcus, Feb 08 2025

from sympy import isprime
from itertools import count
def a(n): return next(k for k in count(2) if k%10 in {3,5,7} and isprime(k**(2*3**n)-6))

a(7) from Michael S. Branicky, Feb 24 2025

a(8) from Georg Grasegger, Apr 17 2025

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A137270 Primes p such that p^2 - 6 is also prime.

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A028880 Primes of the form n^2 - 6.

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A382246 Smallest number k such that k^n - 6 is prime.

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A309726 Numbers k such that k^2 - 12 is prime.

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Mathematica

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A380905 Smallest number k such that k^(2*3^n) - 6 is prime.

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Author

Keywords

Comments

Examples

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PARI

Python

Extensions