# [seqfan] Re: A137921 & "divisor Islands"

franktaw at netscape.net franktaw at netscape.net
Sun Feb 7 21:47:00 CET 2010

```Well, pretty obviously, the intended definition is "Number of proper
divisors d of n such that d+1 is not a divisor of n."  Except that a(1)
should be 0, not 1.

If we just changed "divisor" to "proper divisor": "Number of proper
divisors d of n such that d+1 is not a proper divisor of n.", then we
would have not only a(1) = 0, but also a(2) = 1.

For A175102, we have:

1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 2, 3, 2, 3, 4, 4, 2, 4, 2, 4, 4, 3, 2, 5,
3, 3, 4, 5, 2, 5, 2, 5, 4, 3, 4, 6, 2, 3, 4, 6, 2, 5, 2, 5, 6, 3, 2, 7,
3, 5, 4, 5, 2, 6, 4, 6, 4, 3, 2, 7, 2, 3, 6, 6, 4, 6, 2, 5, 4, 7, 2, 8,
2, 3, 6, 5, 4, 6, 2, 8, 5, 3, 2, 8, 4, 3, 4, 7, 2, 8, 4, 5, 4, 3, 4, 9,
2, 5, 6, 7, 2, 6, 2, 7, 8, 3, 2, 9, 2, 6, 4, 8, 2, 6, 4, 5, 6, 3, 4, 11

using PARI:

a(n)=local(r,ds);ds=divisors(n);r=1;for(k=1,#ds-1,r+=ds[k]+1!=ds[k+1]);r

Change the "r=1" to "r=0" to compute A137921 (with a(1)=0).

-----Original Message-----
From: Leroy Quet <q1qq2qqq3qqqq at yahoo.com>

Regarding A137921: First, by d, does the author mean only PROPER
divisors?
Are there other sequences in the database where "proper divisor" should
have
been included in the title or in a comment, instead of simply "divisor"
(so as
to avoid confusion)?

By the way, with "divisor" of n interpreted to include n itself, then
we have
A137921 equal to this sequence I just submitted (but not yet appearing):

%I A175102
%S A175102 1,1,2,2,2,2,2,3,3,3,2,3,2,3,4,4,2,4,2,4,4
%N A175102 a(n) = number of "divisor islands" of n. A divisor island is
any set
of consecutive divisors of a number where no pairs of consecutive
divisors in
the set are separated by 2 or more.
%e A175102 The divisors of 30 are 1,2,3,5,6,10,15,30. The divisor
islands are
(1,2,3), (5,6), (10), (15), (30). (Note that the differences between
consecutive
divisors 5-3, 10-6, 15-10, and 30-15 are all > 1.) There are 5 such
islands, so
a(30)=5.
%K A175102 more,nonn
%O A175102 1,3

I should have included a cross-ref to A137921, but I did not want to do
that
until the title of A137921 is fixed.

Thanks,
Leroy Quet

[ ( [ ([( [ ( ([[o0Oo0Ooo0Oo(0)oO0ooO0oO0o]]) ) ] )]) ] ) ]

_______________________________________________

Seqfan Mailing list - http://list.seqfan.eu/

```