A384530 Intersection of A055932 and A014574.

4, 6, 12, 18, 30, 60, 72, 108, 150, 180, 192, 240, 270, 420, 432, 600, 810, 1050, 1152, 1620, 2310, 2592, 3000, 3360, 4050, 4800, 5880, 6300, 7350, 7560, 8820, 9000, 9240, 9720, 10500, 11550, 15360, 21600, 23040, 25410, 26250, 26880, 28350, 29400, 30870, 33600

Offset: 1

Author

Ken Clements, Jun 01 2025

Keywords

• nonn

Comments

Numbers that have a complete set of prime factors from 2 to their greatest prime factor (A055932) that are bracketed by twin primes (in A014574).

The density of twin prime numbers around terms appears to be enhanced, since the blocks of prime factors of a(n) "sweep" out possible low prime factors from a(n)-1 and a(n)+1.

Examples

			4 is a term since 4 = prime(1)# * 2 is in A055932 and 4-1 = 3 and 4+1 = 5 are both prime.
6 is a term since 6 = prime(2)# is in A055932 and 6-1 = 5 and 6+1 = 7 are both prime.
30 is a term since 30 = prime(3)# is in A055932 and 30-1 = 29 and 30+1 = 31 are both prime.
420 is a term since 420 = prime(4)# * 2 is in A055932 and 420-1 = 419 and 420+1 = 421 are both prime.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..3719

Crossrefs

Supersequence of A027856.

Cf. A014574, A055932.

Programs

Maple

q:= n-> andmap(isprime, [n-1, n+1]) and (s-> nops(s)=
        numtheory[pi](max(s)))({ifactors(n)[2][.., 1][]}):
select(q, [$1..40000])[];  # Alois P. Heinz, Jun 01 2025

Python

from sympy import factorint, prime, isprime
def is_pi_complete(n):  # n is a term of A055932
    if n <= 1:
        return False
    factors = factorint(n)
    primes = list(factors.keys())
    max_prime, r = max(primes), len(primes)
    return max_prime == prime(r)
def is_twin_prime_bracketed(n):  # n is a term of A014574
    return isprime(n-1) and isprime(n+1)
def aupto(limit):
    return [n for n in range(4, limit+1, 2) if is_twin_prime_bracketed(n) and is_pi_complete(n)]
print(aupto(40_000))