[seqfan] Re: a(n) divides A000201(a(n))^m

[Sean A. Irvine]

Paul,

Below are what I get for the first 100 terms.  I concur with your values.

If n = p_1^{e_1} p_2^{e_2} ... p_r^{e_r}, then require
p_1p_2...p_r | A000201(a(n)).  Thus value for SEQ.(3) is certainly
below max{e_1,e_2,...e_r} at each point.


         SEQ.(1) SEQ.(2) SEQ.(3)
1       1       1       1
2       4       6       2
3       8       12      2
4       25      40      2
5       50      80      2
6       108     174     3
7       169     273     2
8       243     393     5
9       256     414     8
10      338     546     2
11      486     786     5
12      512     828     5
13      729     1179    3
14      768     1242    8
15      972     1572    5
16      1024    1656    4
17      1156    1870    2
18      1215    1965    5
19      2312    3740    2
20      3375    5460    3
21      5000    8090    4
22      7921    12816   2
23      8192    13254   13
24      8748    14154   7
25      10000   16180   4
26      12800   20710   9
27      15000   24270   4
28      15842   25632   2
29      20000   32360   4
30      25000   40450   3
31      50176   81186   10
32      54289   87841   2
33      85184   137830  6
34      88209   142725  6
35      100352  162372  6
36      104976  169854  8
37      108578  175682  2
38      131072  212078  17
39      176418  285450  6
40      177147  286629  11
41      248832  402618  10
42      264627  428175  4
43      320000  517770  9
44      350464  567062  8
45      352836  570900  6
46      366025  592240  4
47      372100  602070  2
48      441045  713625  6
49      460800  745590  11
50      472392  764346  10
51      529254  856350  4
52      744200  1204140 2
53      820125  1326990 8
54      823543  1332520 7
55      900000  1456230 5
56      921600  1491180 6
57      944784  1528692 10
58      1382400 2236770 11
59      1417176 2293038 6
60      1556068 2517770 3
61      1577536 2552506 6
62      1640250 2653980 8
63      1843200 2982360 5
64      1854944 3001362 5
65      2088025 3378495 4
66      2109375 3413040 7
67      2175625 3520235 4
68      2304000 3727950 11
69      2460375 3980970 5
70      2550409 4126648 2
71      2764800 4473540 6
72      3225600 5219130 11
73      3280500 5307960 8
74      3341637 5406882 5
75      3686400 5964720 4
76      3709476 6002058 4
77      3709888 6002724 3
78      4100625 6634950 8
79      4147200 6710310 11
80      4176050 6756990 4
81      4194304 6786526 22
82      4218750 6826080 7
83      4351250 7040470 4
84      4608000 7455900 6
85      4920750 7961940 5
86      5068800 8201490 11
87      5100818 8253296 2
88      5529600 8947080 5
89      5740875 9288930 8
90      5990400 9692670 11
91      6451200 10438260        6
92      6526875 10560705        4
93      6683274 10813764        5
94      6912000 11183850        11
95      7372800 11929440        3
96      7418952 12004116        4
97      7812500 12640890        9
98      8388608 13573052        12
99      8680203 14044863        11
100     8702500 14080940        4

Sean.


Paul D Hanna wrote:
> SeqFans, 
>       Would someone extend/verify these sequences?  
> Below, a(n) refers to the terms in SEQ.1. 
>  
> SEQ. (1)  
> a(n) divides A000201(a(n))^m for some integer m>0, where A000201 is the lower Wythoff sequence. 
>  
> 1,4,8,25,50,108,169,243,256,338,486,512,729,768,972,1024,1156,1215,
> 2312,3375,5000,7921,8192,8748,10000,12800,15000,15842,20000,25000,
>  
> Many of these terms are powers of fibonacci numbers. 
> Perhaps this is expected, since these involve floor(a(n)*phi). 
>   
> SEQ. (2) 
> A000201(a(n))
>  
> 1,6,12,40,80,174,273,393,414,546,786,828,1179,1242,1572,1656,1870,1965,
> 3740,5460,8090,12816,13254,14154,16180,20710,24270,25632,32360,40450,
>   
> SEQ. (3) 
> Least m for which a(n) divides A000201(a(n))^m. 
>  
> 1,2,2,2,2,3,2,5,8,2,5,5,3,8,5,4,2,5,2,3,4,2,13,7,4,9,4,2,4,3,
>  
> Question: is m always <= n ? 
>  
> Thanks, 
>      Paul 
> 
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