A134322 -id:A134322

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A134321

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.

+10
4

8, 10, 14, 18, 22, 24, 26, 34, 38, 40, 46, 56, 58, 60, 62, 72, 74, 82, 84, 86, 94, 106, 110, 118, 122, 132, 134, 142, 146, 156, 158, 166, 178, 182, 194, 202, 206, 210, 214, 218, 220, 226, 254, 262, 274, 278, 298, 302, 314, 326, 334, 346, 358, 362

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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.

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.

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.

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)

LINKS

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

EXAMPLE

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.

MAPLE

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

MATHEMATICA

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 *)

CROSSREFS

Cf. A133779, A134320, A134322, A243932.

KEYWORD

nonn

AUTHOR

Leroy Quet, Oct 20 2007

EXTENSIONS

More terms from Emeric Deutsch and Hugo van der Sanden, Oct 24 2007

STATUS

approved

A134320

Positive integers with more non-isolated divisors than isolated divisors.

+10
2

2, 4, 6, 12, 20, 30, 42, 90

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

A divisor k of n is isolated if neither k-1 nor k+1 divides n (see A133779, A132881).

Is this sequence finite? One can show that, with the exception of a(2) = 4, all terms of this sequence must be of the form m*(m+1), oblong numbers, A002378.

Comments from Hugo van der Sanden, Oct 30 2007 and Oct 31 2007: (Start) A quick program to check found no other example up to 3e6, which certainly suggests it is not just finite but complete.

Partial proof: if adjacent integers k, k+1 both divide n then since they are coprime we also have that k(k+1) divides n, so k < sqrt(n).

I.e. the largest non-isolated factor a number can have is ceiling(sqrt(n)).

Since the divisors are symmetrically disposed around the square root, we have: if n is nonsquare, to be in this sequence it must be an oblong number, with all divisors below the square root non-isolated; if n is square, say n = m^2, then we have n divisible by m^2(m-1), so we require m-1 = 1.

So the only square entry is n = 4.

It remains to prove that there is no oblong number greater than 9*10 that avoids isolated divisors below the square root. (End)