A358665 -id:A358665 - cp's OEIS frontend

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

Tanya Khovanova and Massimo Kofler, Nov 25 2022

nonn

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.

			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.

Cf. A001358, A358657, A358665.

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} &]

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

A358686 Numbers sandwiched between two semiprimes, one of which is a square.

Original entry on oeis.org

5, 50, 120, 122, 288, 290, 528, 842, 960, 1370, 1680, 1850, 2808, 2810, 4488, 5328, 5330, 6240, 6242, 6888, 6890, 9408, 9410, 11880, 12768, 18770, 22200, 22800, 26568, 27888, 36482, 38808, 39600, 52440, 54290, 58080, 63000, 63002, 69170, 72360, 72362, 73442, 76730, 78960

Offset: 1

Tanya Khovanova, Nov 26 2022

nonn

Numbers in A124936 but not in A358665.
All numbers except 5 (the first term) are even.
Subsequence of A124936.

			5 is sandwiched between two semiprimes 4 = 2*2 and 6 = 3*2, one of which is a square. Thus, 5 is in this sequence.
34 is sandwiched between squarefree semiprimes 33 = 3*11 and 35 = 5*7. Thus, 34 is not in this sequence.

Harvey P. Dale, Table of n, a(n) for n = 1..292 (all terms up to 10 million)

Cf. A001358, A006881, A124936, A358665.

Select[Range[100000], Total[Transpose[FactorInteger[# - 1]][[2]]] == 2 && Total[Transpose[FactorInteger[# + 1]][[2]]] == 2 && ! (Transpose[FactorInteger[# - 1]][[2]] == {1, 1} && Transpose[FactorInteger[# + 1]][[2]] == {1, 1}) &]
Mean/@Select[SequencePosition[PrimeOmega[Range[80000]],{2,,2}],AnyTrue[Sqrt[#],IntegerQ]&] (* _Harvey P. Dale, Jun 14 2023 *)

from sympy import factorint
from itertools import count, islice
def agen(): # generator of terms
    nxt = []
    yield 5
    for k in count(6, 2):
        prv, nxt = nxt, list(factorint(k+1).values())
        if (prv==[1, 1] and nxt==[2]) or (prv==[2] and nxt==[1, 1]): yield k
print(list(islice(agen(), 44))) # Michael S. Branicky, Nov 26 2022

cp's OEIS Frontend

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

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