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

[Roland Bacher]

The second sequence A084251 is very close to the famous sequence 
giving the number of partitions of integers 

ID Number: A000041 (Formerly M0663 and N0244)
Sequence:  1,1,2,3,5,7,11,15,22,30,42,56,77,101,135,176,231,297,385,
           490,627,792,1002,1255,1575,1958,2436,3010,3718,4565,5604,
           6842,8349,10143,12310,14883,17977,21637,26015,31185,37338,
           44583,53174,63261,75175,89134
Name:      a(n) = number of partitions of n (the partition numbers).


is this a mere coincidence?    Roland Bacher







> 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
> ...