Sequence A084250: a(2^n)=2^(n+1)-1?

Paul D Hanna pauldhanna at juno.com
Thu May 22 07:48:07 CEST 2003

```Consider the new sequences A084250 and A084251 given below.

Observe that: a(2^n) = 2^(n+1) - 1 (at least for n<=64),
although I can not guess why this should be true.

Could someone give a rationale for this to be true (if so)?

Also, I would appreciate it if someone could extend
these sequences to test the a(2^n) conjecture.

Thanks Much,
Paul
--------------------------------------------------------
ID Number: A084250.

Least distinct positive integers such that

exp(sum(n>=1,a(n)*x^n/n))

yields an integer power series (A084251), where a(1)=1.

Conjecture: a(2^n) = 2^(n+1) - 1.

A084250 is a permutation of the natural numbers:

1,   3,  4,  7,  6, 12,  8, 15, 13, 18,
23, 16, 14, 10,  9, 31, 35, 21, 20,  2,
11, 25, 24, 48, 56, 42, 40, 70, 30, 27,
32, 63, 26, 37, 83, 61, 38, 22, 17, 50,
124,19, 44, 29,108, 72, 95, 64, 57, 68,
89, 46,107,102,138, 78, 80, 90, 60, 71,
62, 34,146,127, 84,100,...
--------------------------------------------------------
ID Number: A084251.

Integer sequence defined by

exp(sum(n>=1,A084250(n)*x^n/n)) = sum(n>=0,A084251(n)*x^n)

where A084250 is the least distinct positive integers
such that A084251(n) is an integer for all n>=0.

A084251 begins:

1,1,2,3,5,7,11,15,22,30,42,
57,77,102,135,176,230,297,381,486,616,
777,976,1219,1517,1880,2320,2854,3499,4273,5203,
6315,7645,9228,11111,13344,15987,19106,22786,27113,32197,
38158,45132,53283,62793,73871,86754,101718,119069,139170,162416,
189276,220261,255969,297062,344308,398558,460794,532099,613722,707054,
813671,935344,1074072,1232086,1411912,1616377,...

Example.

A(x) = exp(x + 3x^2/2 + 4x^3/3 + 7x^4/4 + 6x^5/5 + 12x^6/6 +...)
= 1 + 1x + 2x^2 + 3x^3 + 5x^4 + 7x^5 + 11x^6 +...
--------------------------------------------------------

Table of the two sequences:

n A084250 A084251
-- ------- -------
0.   _          1

1.   1          1

2.   3          2
3.   4          3

4.   7          5
5.   6          7
6.  12         11
7.   8         15

8.  15         22
9.  13         30
10.  18         42
11.  23         57
12.  16         77
13.  14        102
14.  10        135
15.   9        176

16.  31        230
17.  35        297
18.  21        381
19.  20        486
20.   2        616
21.  11        777
22.  25        976
23.  24       1219
24.  48       1517
25.  56       1880
26.  42       2320
27.  40       2854
28.  70       3499
29.  30       4273
30.  27       5203
31.  32       6315

32.  63       7645
33.  26       9228
34.  37      11111
35.  83      13344
36.  61      15987
37.  38      19106
38.  22      22786
39.  17      27113
40.  50      32197
41. 124      38158
42.  19      45132
43.  44      53283
44.  29      62793
45. 108      73871
46.  72      86754
47.  95     101718
48.  64     119069
49.  57     139170
50.  68     162416
51.  89     189276
52.  46     220261
53. 107     255969
54. 102     297062
55. 138     344308
56.  78     398558
57.  80     460794
58.  90     532099
59.  60     613722
60.  71     707054
61.  62     813671
62.  34     935344
63. 146    1074072

64. 127    1232086
65.  84    1411912
66. 100    1616377
...
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