[math-fun] Unsolved? problem

[Richard Guy]

Thanks for all your interest in
Charles Trigg's sequence.  The
current feeling is that the sequence(s)
diverge sufficiently rapidly that
it's likely that there are infinfinitely
many non-tributary sequences, and
that it's unlikely that anyone will
prove anything.

[has anyone found specimens of
sequences which appear reluctant to
join the main sequence, but which
do merge themselves ??]

Meanwhile, back at the ranch, there
are some equal rights activists
crying for equal time. The rough
description of the sequence is
`add the distinct prime divisors'

There are four interpretations of this.
1 is not considered to be a prime,
although it was not always thus
(Goldbach, D.N.Lehmer,...) and it is
still occasionally a useful fiction
to consider 1 as the zeroth prime.

So we can include or exclude 1
and, if a term is prime, we can
include or exclude it.

(a) we've looked at including 1
but not including the number
itself,  p --> p+1

(b) least interesting, perhaps
is if we include neither, so,
if we hit a prime, we're stuck:

p --> p --> p --> ...

Is there any interest ?

1,1,1,1,...   2,2,2,... 3,3,3,...
4,6,11,11,... 8,10,17,17,...
9,12,17,... 14,23,23,... 15,23,...
16,18,23,... 20,27,30,40,47,47,...
21,31,31,... 22,35,47,...
24,29,29,...  32,34,53,53,...
33,47,... 36,41,... 38,59,...
42,54,59,... 44,57,79,...

E&OE  What's the largest # of distinct
terms that anyone can find?  Perhaps
here it's possible to prove something.

(c)  Include  p  but not  1:

p --> 2p --> 3p+2 --> ...

2,4,6,11,22,35,47,94,143,167,334,503,...
3,6.
5,10,17,34,53,106,161,191,382,575,603,673,...
7,14,23,46,71,142,215,263,526,791,911,...
8,10.
12,17.
13,26,41,82,125,130,150,160,167,...(aha)
15,23.
16,18,23.
19,38,59,118,179,358,539,557,1114,1673,...
20,27,30,40,47.
21,31,62,95,119,143.
24,29,58,89,178,269,538,809,1618,2429,...

(d)  Include  1  and  p  (Cunningham
chains!)  p --> 2p+1 --> ...

1,2,5,11,23,47,95,120,131,263,527,576,582,...
3,7,15,24,30,41,83,167,335,408,431,863,...
4,7.  (period, or aha! indicates tribulation)
6,12,18,24.
8,11.
9,13,27,31,63,74,114,139,279,314,474,559,...
10,18.
14,24.
16,19,39,56,66,83 (aha)
17,35,48,54,60,71,143,168,181,363,378,391,
             432,438,517,576 (aha)
20,28,38,60.
21,32,35.
22,36,42,55,72,78,97,195,217,256,259,304,
             326,492,539,558,595,625,631,...
25,31.
26,42.
29,59,119,144,150,161,192,198,215,264,281,
             563,1127,...

that's enough mistakes and cats amongst
pigeons for today.   Best to all,   R.

On Wed, 13 Apr 2005, Richard Guy wrote:

> Thanks for several responses.
> I've got as far as Math Mag
> 48(1975) 301 and find:
>
> ``C.W.Trigg, C.C.Oursler, and
> R.Cormier & J.L.Selfridge have
> sent calculations on Problem 886
> [Nov 1973] for which we had
> received only partial results
> [Jan 1975].
>
> Given an ... (restatement of
> problem)
>
> C & S sent the following results:
> There appear to be 5 seqs beginning with
> integers less than 1000 which do not merge.
> These sequences were carried out to
> 10^8 or more.  The calculations are:
>
> 1,2,3,4,7,8,..,96532994,144799494,...(31)
> 393,528,545,660,682,727,...,97622612,
>             122028268,... (9)
> 412,518,565,684,709,710,..., 92029059,
>             102254514,... (46)
> 668,838,1260,1278,1355,1632,...,91127590,
>             100240357,... (52)
> 932,1168,1244,1558,1621,1622,...,98457737,
>             112523136,... (30)
>
> The numbers in parens show the numbers
> of terms between 50000000and 10^8.  The
> rate of growth of these sequences suggests that
> there are likely an inf no of mutually
> indep seqs.
>
> [[30 yrs on our computers, human &
> electronic, shd be able to improve on
> this.  Should the last 4 of the above
> 5 seqs be in OEIS ??    R.]]
>
> On Wed, 13 Apr 2005, Richard Guy wrote:
>> 
>> I came across Problem 886,
>> Math Mag 48(1975) 57--58
>> which isn't properly stated
>> but should read as in OEIS
>> A003508 :
>> 
>> a(n) = a(n-1) + 1 + sum of
>> distinct prime factors of
>> a(n-1) that are < a(n-1).