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

1,1

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.

Links

Table of n, a(n) for n=1..43.

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

Cf. A001358, A358657, A358665.

Sequence in context: A165078 A135780 A069701 * A349064 A124144 A276564

Adjacent sequences: A358663 A358664 A358665 * A358667 A358668 A358669

Keyword

nonn

Author

Tanya Khovanova and Massimo Kofler, Nov 25 2022

Status

approved