Harmonic Number Continued Fractions Of Distinct Terms

[Maximilian Hasler]

On Sat, May 31, 2008 at 2:01 PM, Leroy Quet <q1qq2qqq3qqqq at yahoo.com> wrote:
> Consider the continued fractions of the harmonic
> numbers H(n) = sum{k=1 to n} 1/k.
> I wanted to submit the sequence of positive
> integers n such that the simple continued
> fraction of each H(n) contains all distinct
> terms.
> Using only the terms given in A100398, I get that
> the sequence begins: 1,2,4,... (There are no
> other n's <= 15.)
> My instinct is that there are only a finite
> number of terms in this sequence.

a) at the end of the mail, I give the c.f.s up to n=50
b) maybe I'll submit a b-file for A100398, at that occasion the "tabl"
should be corrected to "tabf".
c) there are no more terms of your sequence up to n=8000
d) I think it would be quite difficult to prove that the sequence is
finite, even if it is probable that many heuristic arguments in favour
of this conjecture can be found.

Regards,
Maximilian

(13:58) gp > for(n=1,10^4,n%500|print1("/*"n"*/");#Set(t=contfrac(H(n)))==#t
& print1(n", "))
1, 2, 4, /*500*//*1000*//*1500*//*2000*//*2500*//*3000*//*3500*//*4000*//*4500*//*5000*//*5500*//*6000*//*6500*//*7000*//*7500*//*80
00*/
  *** sum: user interrupt after 9mn, 35,563 ms.
(13:53) gp > for(n=1,50,print([n,H(n),contfrac(H(n))]))
[1, 1, [1]]
[2, 3/2, [1, 2]]
[3, 11/6, [1, 1, 5]]
[4, 25/12, [2, 12]]
[5, 137/60, [2, 3, 1, 1, 8]]
[6, 49/20, [2, 2, 4, 2]]
[7, 363/140, [2, 1, 1, 2, 5, 5]]
[8, 761/280, [2, 1, 2, 1, 1, 5, 7]]
[9, 7129/2520, [2, 1, 4, 1, 5, 1, 1, 7, 1, 3]]
[10, 7381/2520, [2, 1, 13, 12, 1, 3, 1, 2]]
[11, 83711/27720, [3, 50, 3, 4, 6, 1, 5]]
[12, 86021/27720, [3, 9, 1, 2, 4, 1, 1, 1, 15, 4]]
[13, 1145993/360360, [3, 5, 1, 1, 4, 2, 1, 3, 2, 1, 3, 1, 4, 1, 6]]
[14, 1171733/360360, [3, 3, 1, 39, 3, 1, 13, 3, 13]]
[15, 1195757/360360, [3, 3, 7, 43, 1, 1, 1, 17, 7]]
[16, 2436559/720720, [3, 2, 1, 1, 1, 2, 9, 1, 3, 2, 1, 8, 3, 1, 1, 1, 2]]
[17, 42142223/12252240, [3, 2, 3, 1, 1, 1, 2, 1, 14, 1, 1, 8, 2, 4, 1,
2, 1, 11]]
[18, 14274301/4084080, [3, 2, 50, 1, 1, 1, 1, 7, 1, 61, 15]]
[19, 275295799/77597520, [3, 1, 1, 4, 1, 2, 1, 3, 1, 25, 2, 2, 2, 3,
4, 1, 1, 4, 1, 6]]
[20, 55835135/15519504, [3, 1, 1, 2, 17, 3, 2, 1, 2, 9, 1, 6, 7, 2, 2, 2]]
[21, 18858053/5173168, [3, 1, 1, 1, 4, 1, 1, 4, 1, 2, 1, 1, 1, 1, 6, 3, 2, 36]]
[22, 19093197/5173168, [3, 1, 2, 4, 3, 1, 2, 1, 2, 5, 1, 1, 7, 1, 1, 2, 1, 13]]
[23, 444316699/118982864, [3, 1, 2, 1, 3, 4, 2, 1, 2, 4, 2, 5, 1, 1,
3, 1, 2, 3, 1, 2, 3, 1, 2]]
[24, 1347822955/356948592, [3, 1, 3, 2, 6, 2, 1, 15, 11, 1, 1, 1, 1,
3, 2, 1, 1, 17, 6]]
[25, 34052522467/8923714800, [3, 1, 4, 2, 3, 3, 1, 4, 2, 2, 1, 1, 2,
5, 1, 3, 6, 2, 1, 18, 3, 1, 2, 3]]
[26, 34395742267/8923714800, [3, 1, 5, 1, 6, 1, 1, 1, 3, 9, 1, 1, 3,
1, 2, 1, 3, 2, 1, 3, 1, 4, 20, 1, 7]]
[27, 312536252003/80313433200, [3, 1, 8, 4, 1, 2, 3, 2, 1, 2, 5, 1, 1,
1, 3, 68, 5, 3, 6, 1, 1, 4, 1, 3]]
[28, 315404588903/80313433200, [3, 1, 12, 1, 2, 1, 2, 1, 1, 45, 2, 2,
3, 1, 1, 3, 9, 8, 1, 2, 1, 2, 1, 2, 2, 4]]
[29, 9227046511387/2329089562800, [3, 1, 25, 12, 1, 3, 1, 2, 2, 1, 1,
2, 1, 1, 8, 9, 11, 1, 3, 1, 5, 1, 1, 1, 2, 1, 2, 7, 2]]
[30, 9304682830147/2329089562800, [3, 1, 198, 2, 18, 10, 8, 1, 2, 1,
1, 15, 1, 30, 1, 6, 1, 1, 5, 1, 9]]
[31, 290774257297357/72201776446800, [4, 36, 1, 2, 2, 1, 1, 1, 88, 2,
34, 1, 1, 5, 16, 13, 1, 11, 3, 1, 2, 2, 3, 4]]
[32, 586061125622639/144403552893600, [4, 17, 10, 2, 11, 1, 10, 2, 4,
3, 5, 1, 2, 1, 3, 1, 13, 2, 2, 1090, 3, 1, 2]]
[33, 53676090078349/13127595717600, [4, 11, 3, 1, 4, 1, 2, 4, 5, 1, 1,
11, 1, 2, 2, 10, 1, 1, 6, 3, 12, 1, 1, 120, 1, 2]]
[34, 54062195834749/13127595717600, [4, 8, 2, 5, 1, 2, 11, 2, 1, 5, 3,
244, 1, 98, 1, 1, 1, 1, 3, 2, 1, 2, 1, 5, 2]]
[35, 54437269998109/13127595717600, [4, 6, 1, 4, 2, 1, 10, 2, 2, 1,
29, 1, 1, 9, 2, 1, 4, 1, 2, 3, 1, 1, 1, 2, 2, 1, 2, 9, 2, 2, 1,
2]]
[36, 54801925434709/13127595717600, [4, 5, 1, 2, 1, 2, 5, 2, 1, 2, 1,
21, 1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 2, 1, 1, 1, 6, 1, 3, 29, 1,
13, 1, 2]]
[37, 2040798836801833/485721041551200, [4, 4, 1, 24, 2, 2, 1, 1, 14,
1, 1, 1, 16, 1, 1, 7, 1, 14, 1, 1, 5, 2, 15, 2, 3, 1, 2, 1, 2,
1, 3, 1, 6]]
[38, 2053580969474233/485721041551200, [4, 4, 2, 1, 1, 2, 1, 2, 3, 1,
111, 6, 1, 6, 5, 25, 3, 2, 1, 4, 1, 3, 43, 2, 7, 5]]
[39, 2066035355155033/485721041551200, [4, 3, 1, 16, 1, 8, 9, 7, 3, 4,
1, 35, 11, 1, 1, 3, 1, 1, 4, 8, 567, 1, 1, 2]]
[40, 2078178381193813/485721041551200, [4, 3, 1, 1, 2, 3, 1, 1, 1, 4,
3, 2, 1, 1, 13, 1, 1, 7, 4, 22, 16, 1, 1, 2, 8, 1, 3, 1, 8, 13
, 3]]
[41, 85691034670497533/19914562703599200, [4, 3, 3, 3, 9, 6, 5, 2, 7,
1, 2, 7, 2, 2, 1, 1, 4, 1, 10, 2, 1, 1, 14, 1, 3, 20, 1, 5, 3,
 10, 2, 2]]
[42, 12309312989335019/2844937529085600, [4, 3, 16, 1, 1, 9, 6, 3, 2,
37, 6, 6, 13, 1, 1, 24, 4, 1, 4, 1, 1, 1, 1, 1, 3, 6, 1, 1, 1,
 1, 2]]
[43, 532145396070491417/122332313750680800, [4, 2, 1, 6, 1812, 4, 3,
1, 4, 5, 4, 3, 1, 12, 1, 1, 1, 3, 4, 1, 1, 2, 3, 2, 2, 1, 2, 4,
 1, 1, 2, 2, 1, 3, 2, 2]]
[44, 5884182435213075787/1345655451257488800, [4, 2, 1, 2, 6, 2, 59,
2, 4, 2, 25, 1, 1, 2, 3, 1, 9, 1, 2, 1, 1, 1, 1, 2, 33, 10, 2,
3, 4, 1, 10, 1, 1, 40, 1, 2]]
[45, 5914085889685464427/1345655451257488800, [4, 2, 1, 1, 7, 3, 6, 1,
5, 7, 3, 14, 2, 4, 1, 3, 1, 41, 1, 1, 1, 2, 1, 2, 4, 3, 3, 2,
 1, 2, 3, 3, 2, 18, 1, 1, 1, 1, 11]]
[46, 5943339269060627227/1345655451257488800, [4, 2, 2, 1, 1, 336, 1,
6, 2, 1, 2, 14, 1, 2, 32, 59, 2, 19, 4, 13, 1, 1, 1, 2, 1, 1,
1, 3, 4, 1, 16, 3]]
[47, 280682601097106968469/63245806209101973600, [4, 2, 3, 1, 1, 7, 1,
6, 10, 1, 3, 6, 2, 15, 1, 1, 3, 2, 1, 5, 4, 121, 1, 23, 2, 2,
 61, 4, 1, 3, 1, 1, 5, 59]]
[48, 282000222059796592919/63245806209101973600, [4, 2, 5, 1, 1, 3, 4,
1, 31, 1, 3, 29, 1, 1, 32, 2, 1, 1, 1, 4, 2, 8, 1, 1, 2, 1, 1
, 2, 7, 1, 8, 1, 5, 1, 17, 1, 4, 2, 5, 1, 3, 8]]
[49, 13881256687139135026631/3099044504245996706400, [4, 2, 11, 1, 1,
10, 1, 2, 2, 1, 3, 5, 2, 1, 4, 1, 3, 2, 9, 2, 1, 4, 2, 5, 3, 2
4, 16, 5, 37, 1, 3, 1, 1, 1, 1, 1, 4, 1, 1, 6, 2, 1, 1, 2, 2, 2, 2]]
[50, 13943237577224054960759/3099044504245996706400, [4, 2, 314, 10,
14, 19, 1, 1, 1, 5, 1, 1, 11, 1, 1, 1, 1, 1, 2, 1, 19, 2, 1, 1,
 1, 1, 9, 1, 3, 1, 5, 2, 55, 2, 1, 2, 1, 1, 2, 2, 3, 6, 1, 4]]