Proximal Fifth Powers That Are Closer To A Cube Than To A Square

[Gerald McGarvey]

For the entries listed for this sequence, the continued fractions of 
a(n)^(5/3)
(which are all close to integers) look interesting.  PARI/GP code:
M=[199,1354,4995,7320,7994,12634,44217,91116,177682, 
394826,458908,462763,512012]
for(i=1,13,print(contfrac(M[i]^(5/3))))
result with precision at \p 38:

[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]
[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]
[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]
[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]
[3196001, 11996, 1, 3331, 4, 1, 1, 120, 1, 1, 1, 1, 2, 2, 1, 130, 6, 6]
[6853327, 23453, 1, 5, 2, 1, 18, 1, 2, 2, 4, 1, 7, 4, 1, 264, 2, 6, 7, 1, 
1, 1, 2]
[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]
[184497751, 91121, 1, 25310, 2, 1, 1, 2, 1, 229, 2, 1, 1, 1, 1, 1, 3]
[561572359, 1724744, 1, 5, 1, 1, 1, 6, 7, 99, 1, 1, 1, 28]
[2124921577, 1, 10937708, 1, 1, 7, 5, 1, 9, 1, 1, 3, 1, 9, 1, 14, 5]
[2730293789, 228003, 1, 1, 4, 2, 1, 7, 3, 1, 1, 4, 4, 1, 1, 1, 4, 4, 6]
[2768626553, 268090, 1, 5, 12, 2, 1, 1, 2, 4, 2, 1, 9, 3, 1, 4, 2, 1, 28]
[3276928000, 1, 384001, 1, 106666, 1, 3, 2]

Note regarding some of the terms:
11996 / 3331 ~  3.60132092464725
91121 / 25310 ~  3.60019755037535
384001 / 106666 ~  3.60003187519922
the corresponding a(n)'s and round(a(n)^(5/3))'s and their factors:
7994            3196001 3196001
91116           184497751       184497751
512012  3276928001      8831 * 371071

for the proximal members noted:
N=[14706104,14706146,66430098,66430152]
for(i=1,4,print(contfrac(N[i]^(5/3))))
result with precision at \p 66:

[882733052251, 6302621, 1, 1750727, 18, 15915, 1, 1, 1, 7, 1, 2, 2, 17188, 
1, 1, 5, 1, 6, 5]
[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]

[10896193872001, 22143371, 1, 6150935, 2, 1, 1, 2, 1, 55916, 1, 7, 2, 1, 1, 
1, 2, 60415]
[10896208634250, 1, 22143376, 1, 6150937, 1, 1, 1, 1, 3, 55917, 2, 1, 7, 1, 
2, 1, 1, 60355]

the rounded values of N[i]^(5/3) and their factors:

882733052251             19 * 46459634329
882737254001             22111 * 39922991

10896193872001   10896193872001
10896208634251   29201 * 373145051

the square roots of N[i]^(5/3):

939538.744411852341870556097433
939540.980479829841808017437261
(difference is about 2.236067977499937461339828)

3300938.33205060042667325416685
3300940.56811857792647492147264
(difference is about 2.23606797749980166730579)

sqrt(5) = 2.2360679774997896964091736687....

Also regarding some terms in the continued fractions:

6302621 / 1750727 ~ 3.60000217052687
6302626 / 1750729 ~3.60000091390501
22143371 / 6150935 ~3.60000081288455
22143376 / 6150937 ~3.6000004552152

and there's also other interesting ratios of the terms.

Also:
7994^(1/3) = 19.9949987494789061066872322273
91116^(1/3) = 44.9985184697427892919942820484
512012^(1/3) = 80.000624995117251077294366744
14706104^(1/3) = 244.999883381868688918513801323
14706146^(1/3) = 245.00011661802029250512935941
66430098^(1/3) = 404.999945130308066900792364781
66430152^(1/3) = 405.000054869677065532407366289

The rounded values, 20, 45, 80, 245, and 405 are all in A033429 (5*n^2) 
which begins as
5, 20, 45, 80, 125, 180, 245, 320, 405, 500, 605, 720, 845, 980,

It's looking less like coincidence.
Regards,
Gerald

At 11:43 AM 4/9/2006, Hans Havermann wrote:
>More specifically, Ed Pegg's A117594, where numbers which are cubes
>themselves are trivially excluded. When I first noticed the relative
>proximity of 14706104 and 14706146, which are both members of this
>sequence, I wondered if this was coincidental. I've now run across
>another example: 66430098 and 66430152.
>
>Might there be a number-theoretic explanation?
-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://list.seqfan.eu/pipermail/seqfan/attachments/20060409/1deec0a8/attachment-0001.htm>