A132895 Even numbers for which all divisors, with the exception of 1 and 2, are isolated. A positive divisor, k, of n is isolated if neither (k-1) nor (k+1) divides n.

2, 4, 8, 10, 14, 16, 22, 26, 28, 32, 34, 38, 44, 46, 50, 52, 58, 62, 64, 68, 70, 74, 76, 82, 86, 88, 92, 94, 98, 104, 106, 116, 118, 122, 124, 128, 130, 134, 136, 142, 146, 148, 152, 154, 158, 164, 166, 170, 172, 176, 178, 184, 188, 190, 194, 196, 202, 206, 208, 212

Offset: 1

Author

Emeric Deutsch, Oct 16 2007, Oct 19 2007

Keywords

• nonn

Comments

Obviously, all divisors of an odd number are isolated.

The numbers of terms that do not exceed 10^k, for k = 1, 2, ..., are 4, 29, 278, 2782, 27813, 278055, 2780548, 27805234, 278052138, 2780519314, ... . Apparently, the asymptotic density of this sequence exists and equals 0.27805... . - Amiram Eldar, Apr 20 2025

Examples

			28 is a term of the sequence because its divisors are 1, 2, 4, 7, 14, 28 and only 1 and 2 are non-isolated.
30 does not belong to the sequence because its divisors are 1, 2, 3, 4, 6, 8, 12, 24 and 1, 2, 3, 4 are non-isolated.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A112886, A133950, A133951, A088722-A088726.

Programs

Maple

with(numtheory): b:=proc(n) local div,ISO,i: div:=divisors(n): ISO:={}: for i to tau(n) do if member(div[i]-1,div)=false and member(div[i]+1,div)=false then ISO:=`union`(ISO,{div[i]}) end if end do end proc: a:=proc(n) if nops(b(n))= tau(n)-2 then n else end if end proc: seq(a(n), n=4..200);

Mathematica

Select[2*Range[120],Min[Differences[Rest[Divisors[#]]]]>1&] (* Harvey P. Dale, Jul 13 2022 *)

PARI

isok(k) = if(k%2, 0, if(!(k%3), 0, fordiv(k, d, if(d > 1 && !(k % (d+1)), return(0))); 1)); \\ Amiram Eldar, Apr 20 2025

Python

from itertools import count, islice
from sympy import divisors
from sympy.ntheory.primetest import is_square
def A132895_gen(startvalue=2): # generator of terms >= startvalue
    return filter(lambda n:all(d==1 or not is_square((d<<3)+1) for d in divisors(n>>1,generator=True)), count(max(startvalue+(startvalue&1),2),2))
A132895_list = list(islice(A132895_gen(),40)) # Chai Wah Wu, Jun 07 2025

Formula

a(n) = 2*A112886(n). - Ray Chandler, May 29 2008

Extensions

Corrected and extended by Ray Chandler, May 29 2008