A358666 Numbers such that the two numbers before and the two numbers after are squarefree semiprimes.

144, 204, 216, 300, 696, 1140, 1764, 2604, 3240, 3900, 4536, 4764, 5316, 5460, 6000, 6504, 7116, 7836, 7860, 8004, 8484, 9300, 9864, 9936, 10020, 11760, 12180, 13140, 13656, 14256, 15096, 16020, 16440, 16860, 18000, 19536, 20016, 20136, 20280, 21780, 22116, 22236, 23940

Offset: 1

Author

Tanya Khovanova and Massimo Kofler, Nov 25 2022

Keywords

• nonn

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.

Examples

			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.
		

Crossrefs

Cf. A001358, A358657, A358665.

Programs

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} &]

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