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For the entries listed for this sequence, the continued fractions of
a(n)^(5/3) <br>
(which are all close to integers) look interesting. PARI/GP
code:<br>
M=[199,1354,4995,7320,7994,12634,44217,91116,177682,
394826,458908,462763,512012] <br>
for(i=1,13,print(contfrac(M[i]^(5/3)))) <br>
result with precision at \p 38:<br><br>
[6782, 1, 2776, 1, 7, 3, 1, 1, 1, 2, 2, 38, 1, 10, 1, 1, 1, 3, 5, 1, 6,
2, 5, 40, 6, 2, 6]<br>
[165715, 1, 2000, 1, 17, 8, 1, 1, 1, 1, 1, 7, 1, 2, 2, 2, 4, 1, 1, 1, 6,
3, 3, 2, 22, 4, 30, 13]<br>
[1459572, 1, 40565, 1, 11, 4, 1, 1, 1, 30, 6, 2, 2, 1, 2, 2, 1, 2, 1, 2,
1, 2, 1, 3, 2, 1, 1, 3, 1, 3]<br>
[2759635, 1, 29461, 1, 1, 5, 5, 9, 1, 5, 2, 3, 6, 1, 7, 1, 1, 4, 1, 2, 1,
2, 2, 1, 1, 2, 1, 4, 1, 1, 4]<br>
[3196001, 11996, 1, 3331, 4, 1, 1, 120, 1, 1, 1, 1, 2, 2, 1, 130, 6,
6]<br>
[6853327, 23453, 1, 5, 2, 1, 18, 1, 2, 2, 4, 1, 7, 4, 1, 264, 2, 6, 7, 1,
1, 1, 2]<br>
[55290279, 1, 203270, 1, 1, 1, 2, 1, 3, 1, 2, 3, 1, 2, 1, 1, 1, 1, 3, 78,
1, 1, 1, 7, 1, 7, 8]<br>
[184497751, 91121, 1, 25310, 2, 1, 1, 2, 1, 229, 2, 1, 1, 1, 1, 1,
3]<br>
[561572359, 1724744, 1, 5, 1, 1, 1, 6, 7, 99, 1, 1, 1, 28]<br>
[2124921577, 1, 10937708, 1, 1, 7, 5, 1, 9, 1, 1, 3, 1, 9, 1, 14,
5]<br>
[2730293789, 228003, 1, 1, 4, 2, 1, 7, 3, 1, 1, 4, 4, 1, 1, 1, 4, 4,
6]<br>
[2768626553, 268090, 1, 5, 12, 2, 1, 1, 2, 4, 2, 1, 9, 3, 1, 4, 2, 1,
28]<br>
[3276928000, 1, 384001, 1, 106666, 1, 3, 2]<br><br>
Note regarding some of the terms:<br>
11996 / 3331 ~ 3.60132092464725 <br>
91121 / 25310 ~ 3.60019755037535 <br>
384001 / 106666 ~ 3.60003187519922 <br>
the corresponding a(n)'s and round(a(n)^(5/3))'s and their factors:<br>
7994<x-tab> </x-tab><x-tab> </x-tab>3196001<x-tab> </x-tab>3196001
<br>
91116<x-tab> </x-tab><x-tab> </x-tab>184497751<x-tab> </x-tab>184497751
<br>
512012<x-tab> </x-tab>3276928001<x-tab> </x-tab>8831
* 371071 <br><br>
for the proximal members noted:<br>
N=[14706104,14706146,66430098,66430152]<br>
for(i=1,4,print(contfrac(N[i]^(5/3)))) <br>
result with precision at \p 66:<br><br>
[882733052251, 6302621, 1, 1750727, 18, 15915, 1, 1, 1, 7, 1, 2, 2,
17188, 1, 1, 5, 1, 6, 5]<br>
[882737254000, 1, 6302626, 1, 1750729, 3, 1, 1, 1, 1, 15915, 8, 2, 1, 1,
1, 1, 1, 17188, 2, 1, 2, 2, 1, 2]<br><br>
[10896193872001, 22143371, 1, 6150935, 2, 1, 1, 2, 1, 55916, 1, 7, 2, 1,
1, 1, 2, 60415]<br>
[10896208634250, 1, 22143376, 1, 6150937, 1, 1, 1, 1, 3, 55917, 2, 1, 7,
1, 2, 1, 1, 60355]<br><br>
the rounded values of N[i]^(5/3) and their factors:<br><br>
882733052251<x-tab> </x-tab><x-tab> </x-tab>
19 * 46459634329 <br>
882737254001<x-tab> </x-tab><x-tab> </x-tab>
22111 * 39922991 <br><br>
10896193872001<x-tab> </x-tab> 10896193872001 <br>
10896208634251<x-tab> </x-tab> 29201 * 373145051 <br><br>
the square roots of N[i]^(5/3):<br><br>
939538.744411852341870556097433 <br>
939540.980479829841808017437261 <br>
(difference is about 2.236067977499937461339828)<br><br>
3300938.33205060042667325416685 <br>
3300940.56811857792647492147264 <br>
(difference is about 2.23606797749980166730579)<br><br>
sqrt(5) = 2.2360679774997896964091736687....<br><br>
Also regarding some terms in the continued fractions: <br><br>
6302621 / 1750727 ~ 3.60000217052687 <br>
6302626 / 1750729 ~3.60000091390501 <br>
22143371 / 6150935 ~3.60000081288455 <br>
22143376 / 6150937 ~3.6000004552152 <br><br>
and there's also other interesting ratios of the terms.<br><br>
Also:<br>
7994^(1/3) = 19.9949987494789061066872322273 <br>
91116^(1/3) = 44.9985184697427892919942820484 <br>
512012^(1/3) = 80.000624995117251077294366744 <br>
14706104^(1/3) = 244.999883381868688918513801323 <br>
14706146^(1/3) = 245.00011661802029250512935941 <br>
66430098^(1/3) = 404.999945130308066900792364781 <br>
66430152^(1/3) = 405.000054869677065532407366289 <br><br>
The rounded values, 20, 45, 80, 245, and 405 are all in A033429 (5*n^2)
which begins as <br>
<tt>5, 20, 45, 80, 125, 180, 245, 320, 405, 500, 605, 720, 845, 980,</tt>
<br><br>
It's looking less like coincidence.<br>
Regards,<br>
Gerald<br><br>
At 11:43 AM 4/9/2006, Hans Havermann wrote:<br>
<blockquote type=cite class=cite cite="">More specifically, Ed Pegg's A117594, where numbers which are cubes <br>
themselves are trivially excluded. When I first noticed the relative <br>
proximity of 14706104 and 14706146, which are both members of this <br>
sequence, I wondered if this was coincidental. I've now run across <br>
another example: 66430098 and 66430152.<br><br>
Might there be a number-theoretic explanation?</blockquote></body>
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