Re Recaman

[David W. Wilson]

"N. J. A. Sloane" wrote:
> 
> maybe i wasn't clear - we did compute the first billion
> terms this afternoon (a 15 Gig fortran program) and 1355
> still has not appeared.
> NJAS

I don't know where I stand on NJAS's conjecture that we will eventually
find every positive integer in Recaman.  1355 et al seem pretty
recalcitrant.

I plotted R(n)/n for 1 <= n <= 10^8.  This is the result:

R(n)/n  count   plot of count, rounded to nearest 100000.
0.0-0.1  909297 *********
0.1-0.2 2592480 **************************
0.2-0.3 3777613 **************************************
0.3-0.4 3131651 *******************************
0.4-0.5 2385180 ************************
0.5-0.6 3775287 **************************************
0.6-0.7 3562948 ************************************
0.7-0.8 3114124 *******************************
0.8-0.9 2563991 **************************
0.9-1.0 2469097 *************************
1.0-1.1 4613054 **********************************************
1.1-1.2 4983682 **************************************************
1.2-1.3 5020096 **************************************************
1.3-1.4 3858452 ***************************************
1.4-1.5 3003720 ******************************
1.5-1.6 3780634 **************************************
1.6-1.7 3568269 ************************************
1.7-1.8 3128279 *******************************
1.8-1.9 2571816 **************************
1.9-2.0 2479610 *************************
2.0-2.1 4053901 *****************************************
2.1-2.2 3249495 ********************************
2.2-2.3 2413422 ************************
2.3-2.4 1602630 ****************
2.4-2.5 1431604 **************
2.5-2.6 1173315 ************
2.6-2.7 1168098 ************
2.7-2.8  964817 **********
2.8-2.9  876792 *********
2.9-3.0  803576 ********
3.0-3.1  903408 *********
3.1-3.2 1554853 ****************
3.2-3.3 1560822 ****************
3.3-3.4 1546568 ***************
3.4-3.5 1505530 ***************
3.5-3.6 1376777 **************
3.6-3.7 1195223 ************
3.7-3.8 1037599 **********
3.8-3.9 1014149 **********
3.9-4.0  886684 *********
4.0-4.1  620182 ******
4.1-4.2  750023 ********
4.2-4.3  462604 *****
4.3-4.4  733809 *******
4.4-4.5  755930 ********
4.5-4.6  219816 **
4.6-4.7   41077 
4.7-4.8  108644 *
4.8-4.9  158454 **
4.9-5.0   95206 *
5.0-5.1   78195 *
5.1-5.2   55851 *
5.2-5.3   74804 *
5.3-5.4   71553 *
5.4-5.5   66473 *
5.5-5.6   12989 
5.6-5.7    9078 
5.7-5.8   22751 
5.8-5.9   16804 
5.9-6.0    1577 
6.0-6.1   11795 
6.1-6.2    2392 
6.2-6.3    2088 
6.3-6.4    8486 
6.4-6.5    3004 
6.5-6.6    1986 
6.6-6.7     448 
6.7-6.8    1214 
6.8-6.9    3529 
6.9-7.0       0 
7.0-7.1     514 
7.1-7.2       0 
7.2-7.3       0 
7.3-7.4       8 
7.4-7.5       0 
7.5-7.6       1 
7.6-7.7       0 
7.7-7.8     172 


Our smallest Recaman-hard numbers (R(n) = 1355 et al) are near
R(n)/n = 0.  Given that R(n)/n <= 7.3 on such a large range, one might
be tempted to surmise that R(n)/n is bounded.  This would bode well
for NJAS's conjecture, since R(n) would always be only a few steps
away from R(n)/n = 0.

However, if R(n)/n is unbounded, even with very slow average growth,
I would be inclined to think that R(n)/n will eventually drift far
enough from 0 that R(n) will be unlikely to return to some of the small
Recaman-hard numbers.