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

Author

Robert Israel, Apr 01 2024

Keywords

• nonn
• more

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.

Examples

			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.
		

Crossrefs

Cf. A001222. Contained in A371622.

Cf. A115186, A154704.

Programs

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);

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

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

Formula

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

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