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<DIV>Consider the new sequences A084250 and A084251 given below.</DIV>
<DIV> </DIV>
<DIV>Observe that: <STRONG>a(2^n) = 2^(n+1) - 1</STRONG> (at least for
n<=64), <BR>although I can not guess why this should be true.</DIV>
<DIV> </DIV>
<DIV>Could someone give a rationale for this to be true (if so)?</DIV>
<DIV> </DIV>
<DIV>Also, I would appreciate it if someone could extend <BR>these sequences to
test the a(2^n) conjecture.</DIV>
<DIV> </DIV>
<DIV>Thanks Much,<BR>
Paul<BR>--------------------------------------------------------<BR>ID Number:
A084250.</DIV>
<DIV> </DIV>
<DIV><STRONG>Least distinct positive integers such that </STRONG></DIV>
<DIV><STRONG></STRONG> </DIV>
<DIV><STRONG> exp(sum(n>=1,a(n)*x^n/n)) </STRONG></DIV>
<DIV><STRONG></STRONG> </DIV>
<DIV><STRONG>yields an integer power series (A084251), where
a(1)=1.</STRONG></DIV>
<DIV> </DIV>
<DIV><BR>Conjecture: a(2^n) = 2^(n+1) - 1.</DIV>
<DIV> </DIV>
<DIV><BR>A084250 is a permutation of the natural numbers:</DIV>
<DIV> </DIV>
<DIV>1, 3, 4, 7, 6, 12, 8, 15, 13,
18,<BR>23, 16, 14, 10, 9, 31, 35, 21, 20, 2,<BR>11, 25, 24, 48, 56,
42, 40, 70, 30, 27,<BR>32, 63, 26, 37, 83, 61, 38, 22, 17, 50,<BR>124,19, 44,
29,108, 72, 95, 64, 57, 68,<BR>89, 46,107,102,138, 78, 80, 90, 60, 71,<BR>62,
34,146,127,
84,100,...<BR>--------------------------------------------------------<BR>ID
Number: A084251.</DIV>
<DIV> </DIV>
<DIV>Integer sequence defined by</DIV>
<DIV> </DIV>
<DIV><STRONG> exp(sum(n>=1,A084250(n)*x^n/n)) =
sum(n>=0,A084251(n)*x^n)</STRONG></DIV>
<DIV> </DIV>
<DIV>where A084250 is the least distinct positive integers <BR>such that
A084251(n) is an integer for all n>=0.</DIV>
<DIV> </DIV>
<DIV>A084251 begins:</DIV>
<DIV> </DIV>
<DIV>1,1,2,3,5,7,11,15,22,30,42,<BR>57,77,102,135,176,230,297,381,486,616,<BR>777,976,1219,1517,1880,2320,2854,3499,4273,5203,<BR>6315,7645,9228,11111,13344,15987,19106,22786,27113,32197,<BR>38158,45132,53283,62793,73871,86754,101718,119069,139170,162416,<BR>189276,220261,255969,297062,344308,398558,460794,532099,613722,707054,<BR>813671,935344,1074072,1232086,1411912,1616377,...</DIV>
<DIV> </DIV>
<DIV>Example.</DIV>
<DIV> </DIV>
<DIV>A(x) = exp(x + 3x^2/2 + 4x^3/3 + 7x^4/4 + 6x^5/5 + 12x^6/6 +...)</DIV>
<DIV> = 1 + 1x + 2x^2 + 3x^3 + 5x^4 + 7x^5 + 11x^6
+...<BR>--------------------------------------------------------</DIV>
<DIV> </DIV>
<DIV>Table of the two sequences:</DIV>
<DIV> </DIV>
<DIV> n A084250 A084251<BR>-- -------
-------<BR> 0. _
1 </DIV>
<DIV> </DIV>
<DIV> 1.
1 1</DIV>
<DIV> </DIV>
<DIV> 2.
3
2<BR> 3.
4 3</DIV>
<DIV> </DIV>
<DIV> 4.
7
5<BR> 5.
6 7<BR> 6.
12 11<BR> 7.
8 15</DIV>
<DIV> </DIV>
<DIV> 8. 15
22<BR> 9. 13
30<BR>10. 18
42<BR>11. 23
57<BR>12. 16
77<BR>13. 14 102<BR>14.
10 135<BR>15.
9 176</DIV>
<DIV> </DIV>
<DIV>16. 31 230<BR>17.
35 297<BR>18.
21 381<BR>19.
20 486<BR>20.
2 616<BR>21.
11 777<BR>22.
25 976<BR>23.
24 1219<BR>24.
48 1517<BR>25.
56 1880<BR>26.
42 2320<BR>27.
40 2854<BR>28.
70 3499<BR>29.
30 4273<BR>30.
27 5203<BR>31.
32 6315</DIV>
<DIV> </DIV>
<DIV>32. 63 7645<BR>33.
26 9228<BR>34.
37 11111<BR>35.
83 13344<BR>36.
61 15987<BR>37.
38 19106<BR>38.
22 22786<BR>39.
17 27113<BR>40.
50 32197<BR>41. 124
38158<BR>42. 19 45132<BR>43.
44 53283<BR>44.
29 62793<BR>45. 108
73871<BR>46. 72 86754<BR>47.
95 101718<BR>48. 64
119069<BR>49. 57 139170<BR>50.
68 162416<BR>51. 89
189276<BR>52. 46 220261<BR>53.
107 255969<BR>54. 102
297062<BR>55. 138 344308<BR>56.
78 398558<BR>57. 80
460794<BR>58. 90 532099<BR>59.
60 613722<BR>60. 71
707054<BR>61. 62 813671<BR>62.
34 935344<BR>63. 146
1074072</DIV>
<DIV> </DIV>
<DIV>64. 127 1232086<BR>65. 84
1411912<BR>66. 100 1616377<BR>...</DIV></BODY></HTML>