cp's OEIS Frontend

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

%I A358666 #14 Nov 27 2022 10:41:01
%S A358666 144,204,216,300,696,1140,1764,2604,3240,3900,4536,4764,5316,5460,
%T A358666 6000,6504,7116,7836,7860,8004,8484,9300,9864,9936,10020,11760,12180,
%U A358666 13140,13656,14256,15096,16020,16440,16860,18000,19536,20016,20136,20280,21780,22116,22236,23940
%N A358666 Numbers such that the two numbers before and the two numbers after are squarefree semiprimes.
%C A358666 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.
%e A358666 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.
%t A358666 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} &]
%o A358666 (Python)
%o A358666 from itertools import count, islice
%o A358666 from sympy import isprime, factorint
%o A358666 def issfsemiprime(n): return list(factorint(n).values()) == [1, 1] if n&1 else isprime(n//2)
%o A358666 def ok(n): return all(issfsemiprime(n+i) for i in (-2, 2, -1, 1))
%o A358666 def agen(): yield from (k for k in count(12, 12) if ok(k))
%o A358666 print(list(islice(agen(), 43))) # _Michael S. Branicky_, Nov 26 2022
%Y A358666 Cf. A001358, A358657, A358665.
%K A358666 nonn
%O A358666 1,1
%A A358666 _Tanya Khovanova_ and _Massimo Kofler_, Nov 25 2022