A371651 a(n) is the first prime p such that p - 2 and p + 2 both have exactly n prime factors, counted with multiplicity.

5, 23, 173, 2693, 32587, 495637, 4447627, 35303123, 717591877, 928090627, 69692326373, 745041171877, 5012236328123, 64215009765623, 945336806640623, 8885812685546873

Offset

1,1

Comments

a(n) is the first prime p such that A001222(p - 2) = A001222(p + 2) = n.

3*10^9 < a(13) <= 5012236328123.

3*10^9 < a(14) <= 64215009765623.

Links

Table of n, a(n) for n=1..16.

Formula

a(n) > 2*A154704(n) for n > 1.

Example

a(3) = 173 because 173 is prime, 173 - 2 = 171 = 3^2 * 19 and 173 + 2 = 175 = 5^2 * 7 are both products of 3 primes with multiplicity, and no smaller number works.

Maple

V:= Vector(8):
p:= 3: count:= 0:
while count < 8 do
p:= nextprime(p);
i:= numtheory:-bigomega(p-2);
if i <= 8 and V[i] = 0 and numtheory:-bigomega(p+2) = i
     then V[i]:= p; count:= count+1
  fi
od:
convert(V, list);

Prog

(Python)
from sympy import primeomega, nextprime
def A371651(n):
    p = 3
    while True:
        if n == primeomega(p-2) == primeomega(p+2):
            return p
        p = nextprime(p) # Chai Wah Wu, Apr 02 2024
(PARI)
generate(A, B, n) = A=max(A, 2^n); (f(m, p, j) = my(list=List()); if(j==1, forprime(q=max(p, ceil(A/m)), B\m, my(t=m*q); if(isprime(t-2) && bigomega(t-4) == n, listput(list, t-2))), forprime(q = p, sqrtnint(B\m, j), list=concat(list, f(m*q, q, j-1)))); list); vecsort(Vec(f(1, 3, n)));
a(n) = my(x=2^n, y=2*x); while(1, my(v=generate(x, y, n)); if(#v >= 1, return(v[1])); x=y+1; y=2*x); \\ Daniel Suteu, Apr 13 2024

Crossrefs

Cf. A001222. Contained in A371622.

Cf. A115186, A154704.

Sequence in context: A306180 A308443 A116151 * A305127 A332688 A198026

Adjacent sequences: A371648 A371649 A371650 * A371652 A371653 A371654

Keyword

nonn,more

Author

Robert Israel, Apr 01 2024

Extensions

a(11) from Michael S. Branicky, Apr 01 2024

a(12) from Michael S. Branicky, Apr 02 2024

a(13) from Chai Wah Wu, Apr 04 2024

a(14)-a(16) from Daniel Suteu, Apr 13 2024

Status

approved