A179993 -id:A179993 - cp's OEIS frontend

A349327 Primes p such that 2*p^2 is a term of A179993.

2, 3, 7, 13, 43, 127, 211, 293, 743, 757, 797, 811, 1429, 1597, 1721, 2087, 2113, 2239, 2269, 2297, 2381, 2423, 2647, 3079, 3121, 3221, 3863, 4229, 4271, 4957, 5209, 5333, 5923, 6299, 6691, 7127, 7237, 7349, 7757, 7853, 8329, 8513, 8539, 8807, 9127, 9311, 9631, 9661

Offset: 1

Amiram Eldar, Nov 15 2021

nonn

The numbers of the form 2*p^2 where p is a term of this sequence are the only nonsquarefree terms of A179993.

Equivalently, primes p such that p^2 - 2 and 2*p^2 - 1 are also primes, or primes p such that p^2 - 2 is a term of A023204.

			2 is a term since 2*2^2 = 8 = 1*8 = 2*4 is a term of A179993: 8 - 1 = 7 and 4 - 2 = 2 are both primes.
3 is a term since 2*3^2 = 18 = 1*18 = 2*9 = 3*6 is a term of A179993: 18 - 1 = 17, 9 - 2 = 7 and 6 - 3 = 3 are all primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A013929, A023204, A179993.
Intersection of A062326 and A106483.
The prime terms of A225098.

q[n_] := AllTrue[{n, n^2 - 2, 2*n^2 - 1}, PrimeQ]; Select[Range[10000], q]

from itertools import islice
from sympy import isprime, nextprime
def A349327(): # generator of terms
    n = 2
    while True:
        if isprime(n**2-2) and isprime (2*n**2-1): yield n
        n = nextprime(n)
A349327_list = list(islice(A349327(),20)) # Chai Wah Wu, Nov 15 2021

A340728 a(n) is the number of divisors d of n such that n/d - d is prime.

Original entry on oeis.org

0, 0, 1, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 1, 0, 0, 3, 0, 1, 0, 0, 0, 3, 0, 1, 0, 1, 0, 3, 0, 1, 0, 0, 1, 1, 0, 2, 0, 1, 0, 3, 0, 2, 0, 0, 0, 3, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 4, 0, 2, 1, 0, 0, 3, 0, 2, 0, 1, 0, 1, 0, 1, 0, 0, 0, 3, 0, 3, 0, 0, 0, 3, 0, 1, 0, 1, 0, 3, 0, 1, 0, 0, 0, 1, 0, 3, 1

Offset: 1

J. M. Bergot and Robert Israel, Jan 17 2021

nonn

If n is odd, then a(n) = 0 unless n is in A000466, in which case a(n) = 1.

			a(8) = 2; among the divisors {1,2,4,8} of 8, there are two cases where 8/d-d is prime: 8/1-1 = 7 and 8/2-2 = 2.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000466, A010051, A179993, A340729.

f:= proc(n) local D,i,m;
  D:= sort(convert(numtheory:-divisors(n),list));
  m:= nops(D);
  nops(select(i -> isprime(D[m+1-i]-D[i]), [$1..(m+1)/2]));
end proc:
map(f, [$1..100]);

a(n) = sumdiv(n, d, isprime(n/d-d)); \\ Michel Marcus, Jan 18 2021

a(n) = Sum_{d|n} c(n/d-d), where c is the prime characteristic (A010051). - Wesley Ivan Hurt, Jan 18 2021

cp's OEIS Frontend

A349327 Primes p such that 2*p^2 is a term of A179993.

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