OFFSET
1,1
COMMENTS
All numbers in this sequence are divisible by 12. Proof: Suppose n is odd and in this sequence, then either n-1 or n+1 is divisible by 4, creating a contradiction. Suppose n is even, but not divisible by 4, then n-2 is divisible by 4, creating a contradiction. Suppose n is not divisible by 3. Then there exist x such that 3x and 3(x+1) are among squarefree semiprimes surrounding n; one of them is divisible by 6, creating a contradiction.
EXAMPLE
The following numbers are squarefree semiprimes: 214 = 2*107, 215 = 5*43, 217 = 7*31, and 218 = 2*109. Thus, 216 is in this sequence.
MATHEMATICA
Select[Range[100000], Transpose[FactorInteger[# - 2]][[2]] == {1, 1} && Transpose[FactorInteger[# - 1]][[2]] == {1, 1} && Transpose[FactorInteger[# + 2]][[2]] == {1, 1} && Transpose[FactorInteger[# + 1]][[2]] == {1, 1} &]
PROG
(Python)
from itertools import count, islice
from sympy import isprime, factorint
def issfsemiprime(n): return list(factorint(n).values()) == [1, 1] if n&1 else isprime(n//2)
def ok(n): return all(issfsemiprime(n+i) for i in (-2, 2, -1, 1))
def agen(): yield from (k for k in count(12, 12) if ok(k))
print(list(islice(agen(), 43))) # Michael S. Branicky, Nov 26 2022
CROSSREFS
KEYWORD
nonn
AUTHOR
Tanya Khovanova and Massimo Kofler, Nov 25 2022
STATUS
approved