A371622 -id:A371622 - cp's OEIS frontend

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

Robert Israel, Apr 01 2024

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.

			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.

Cf. A001222. Contained in A371622.

Cf. A115186, A154704.

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);

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

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

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

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

A371656 Numbers k such that k - 2 and k + 2 have the same number of prime factors, counted with multiplicity.

Original entry on oeis.org

5, 8, 9, 10, 12, 15, 21, 23, 24, 36, 37, 38, 39, 45, 53, 58, 60, 67, 68, 69, 81, 84, 86, 89, 93, 99, 100, 102, 105, 110, 111, 112, 113, 117, 120, 121, 129, 131, 134, 138, 143, 144, 154, 157, 165, 172, 173, 178, 184, 185, 188, 195, 203, 204, 207, 211, 215, 216, 217, 219, 225, 230, 231, 240, 244

Offset: 1

Robert Israel, Apr 01 2024

nonn

Numbers k such that A001222(k - 2) = A001222(k + 2).

			a(4) = 10 is a term because 10 - 2 = 8 = 2^3 and 10 + 2 = 12 = 2^2 * 3 are both products of 3 primes, counted with multiplicity.

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

Cf. A001222. Contains A371622.

M:= map(numtheory:-bigomega, [$1..10^3]):
select(k -> M[k-2] = M[k+2], [$3 .. 10^3 - 2]);

Select[Range[3,245],PrimeOmega[#-2]==PrimeOmega[#+2]&] (* Stefano Spezia, Apr 01 2024 *)

Suggested by Joerg Arndt

cp's OEIS Frontend

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

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