1,1

Comments from Hugo van der Sanden (hv(AT)crypt.org), 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)

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.

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 (deutsch(AT)duke.poly.edu), Oct 24 2007

Cf. A133779, A134320, A134322.

Adjacent sequences: A134318 A134319 A134320 this_sequence A134322 A134323 A134324

Sequence in context: A100319 A060864 A087695 this_sequence A027693 A100718 A063190

nonn

Leroy Quet (qq-quet(AT)mindspring.com), Oct 20 2007

More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu) and Hugo van der Sanden (hv(AT)crypt.org), Oct 24 2007