%I #19 Jul 16 2014 06:30:37

%S 8,10,14,18,22,24,26,34,38,40,46,56,58,60,62,72,74,82,84,86,94,106,

%T 110,118,122,132,134,142,146,156,158,166,178,182,194,202,206,210,214,

%U 218,220,226,254,262,274,278,298,302,314,326,334,346,358,362

%N Positive integers with the same number of non-isolated divisors as isolated divisors. A divisor k of n is non-isolated if k-1 and/or k+1 also divides n. A divisor k of n is isolated if neither k-1 nor k+1 divides n.

%C Comments from _Hugo van der Sanden_, Oct 30 2007 and Oct 31 2007: (Start) Almost all the entries are of the form 2p or 2pq where q = 2p +/- 1 (and so p is in A005383 or A005384). The exceptions are: 8 18 24 40 56 60 72 84 132 156 210 220 380 ... with no others up to 2e6, suggesting that this exception list is finite and complete.

%C See also my comments on A134320. For the present sequence, we see that elements cannot be perfect squares since those have an odd number of divisors.

%C Thus they must either be oblong numbers with one isolated divisor below the square root (such as the isolated 5 for 110) or non-oblong numbers with all divisors below the square root being non-isolated.

%C I expect that proving this sequence consists only of the two general classes and the finite, complete list of exceptions describe above is also possible and would use a similar approach to the first case. (End)

%H Jens Kruse Andersen, <a href="/A134321/b134321.txt">Table of n, a(n) for n = 1..10000</a>

%e The divisors of 40 are 1,2,4,5,8,10,20,40. Of these, 1,2,4,5 are non-isolated divisors and 8,10,20,40 are isolated divisors. There are the same number of non-isolated divisors (4 in number) as isolated divisors (4 in number), so 40 is in the sequence.

%p with(numtheory): a:=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: nops(ISO) end proc: b:=proc(n) if a(n)=tau(n)-a(n) then n else end if end proc: seq(b(n),n=1..300); # _Emeric Deutsch_, Oct 24 2007

%t fQ[n_] := Block[{d = Divisors@ n}, Length@ d == 2Length@ Select[d, MemberQ[d, # + 1] || MemberQ[d, # - 1] &]]; Select[ Range@ 400, fQ] (* _Robert G. Wilson v_, Jun 22 2014 *)

%Y Cf. A133779, A134320, A134322, A243932.

%K nonn

%O 1,1

%A _Leroy Quet_, Oct 20 2007

%E More terms from _Emeric Deutsch_ and _Hugo van der Sanden_, Oct 24 2007