cp's OEIS Frontend

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.

%I A132895 #20 Jun 07 2025 19:46:15
%S A132895 2,4,8,10,14,16,22,26,28,32,34,38,44,46,50,52,58,62,64,68,70,74,76,82,
%T A132895 86,88,92,94,98,104,106,116,118,122,124,128,130,134,136,142,146,148,
%U A132895 152,154,158,164,166,170,172,176,178,184,188,190,194,196,202,206,208,212
%N 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.
%C A132895 Obviously, all divisors of an odd number are isolated.
%C A132895 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
%H A132895 Amiram Eldar, <a href="/A132895/b132895.txt">Table of n, a(n) for n = 1..10000</a>
%F A132895 a(n) = 2*A112886(n). - _Ray Chandler_, May 29 2008
%e A132895 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.
%e A132895 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.
%p A132895 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);
%t A132895 Select[2*Range[120],Min[Differences[Rest[Divisors[#]]]]>1&] (* _Harvey P. Dale_, Jul 13 2022 *)
%o A132895 (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
%o A132895 (Python)
%o A132895 from itertools import count, islice
%o A132895 from sympy import divisors
%o A132895 from sympy.ntheory.primetest import is_square
%o A132895 def A132895_gen(startvalue=2): # generator of terms >= startvalue
%o A132895     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))
%o A132895 A132895_list = list(islice(A132895_gen(),40)) # _Chai Wah Wu_, Jun 07 2025
%Y A132895 Cf. A112886, A133950, A133951, A088722-A088726.
%K A132895 nonn
%O A132895 1,1
%A A132895 _Emeric Deutsch_, Oct 16 2007, Oct 19 2007
%E A132895 Corrected and extended by _Ray Chandler_, May 29 2008