Finite(?) Sequence (involving divisors)

[Leroy Quet]

Here is a seemingly dumb question.

Is sequence A134320 finite?

Here is the info on this sequence, for those too lazy
to check the EIS:

%I A134320
%S A134320 2,4,6,12,20,30,42,90
%N A134320 Positive integers with more non-isolated
divisors than 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 A134320 Is this sequence finite?

With the exception of a(2) = 4, all terms of this
sequence are of the form m*(m+1). (Oblong numbers:
A002378)
%e A134320 The divisors of 42 are 1,2,3,6,7,14,21,42.
Of these, 1,2,3,6,7 are non-isolated divisors, and
14,21,42 are isolated divisors. There are more
non-isolated divisors (5 in number) than isolated
divisors (3 in number), so 42 is in the sequence.
%Y A134320 A134321,A134322
%O A134320 1
%K A134320 ,more,nonn,

Thanks,
Leroy Quet


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Leroy Quet <q1qq2qqq3qqqq at yahoo.com> wrote:
:Here is a seemingly dumb question.
:
:Is sequence A134320 finite?
:
:Here is the info on this sequence, for those too lazy
:to check the EIS:
:
:%I A134320
:%S A134320 2,4,6,12,20,30,42,90
:%N A134320 Positive integers with more non-isolated
:divisors than 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 A134320 Is this sequence finite?
:
:With the exception of a(2) = 4, all terms of this
:sequence are of the form m*(m+1). (Oblong numbers:
:A002378)
:%e A134320 The divisors of 42 are 1,2,3,6,7,14,21,42.
:Of these, 1,2,3,6,7 are non-isolated divisors, and
:14,21,42 are isolated divisors. There are more
:non-isolated divisors (5 in number) than isolated
:divisors (3 in number), so 42 is in the sequence.
:%Y A134320 A134321,A134322
:%O A134320 1
:%K A134320 ,more,nonn,

A quick program to check found no other example up to 3e6, which
certainly suggests it is not just finite but complete.

Using the same program to find more terms for A134321 "positive integers
with the same number of non-isolated divisors as isolated divisors" found
them much denser:
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
220 226 254 262 274 278 298 302 314 326 334 346 358 362 380 382 386 394 398
422 446 454 458 466 478 482 502 506 514 526 538 542 554 562 566 586 614 622
626 634 662 674 694 698 706 718 734 746 758 766 778 794 802 818 838 842 862
866 878 886 898 914 922 926 934 958 974 982 998 [...]

Almost all these entries are of the form 2p or 2pq where q = 2p +/- 1
(and so p is in A005383 or A005384). The exceptions are:
.. with no other up to 2e6, suggesting the exception list is also finite
and complete.

Hugo




Dear Seqfans,  This week the OEIS will be moved to our
new machine, with a linux rather than unix operating system.
This should not affect the OEIS web pages, but the
email servers, sequences at research.att.com and
Let me know if you see anything supsicious.

My colleague David Applegate has been very helpful in
making the move, by the way.

Neil




(Obviously I can't yet send email from the new machine!)


Dear Seqfans,  This week the OEIS will be moved to our
new machine, with a linux rather than unix operating system.
This should not affect the OEIS web pages, but the
email servers, sequences at research.att.com and
Let me know if you see anything supsicious.

My colleague David Applegate has been very helpful in
making the move, by the way.

Neil