[seqfan] Ordo sequences

[Tomasz Ordowski]

Dear SeqFans!

Let's define the recursion:

a(n+1) is the smallest number > a(n) such that 2^(a(n+1)-1) == 1 (mod a(n)).

Consider the ordo sequences given by this recursion with a fixed initial
term a(1).

Conjecture: For any initial term, this sequence is finite (ends with an
even term).

For example: 3, 5, 9, 13, 25, 41, 61, 121, 221, 241, 265, 313, 469, 529,
and 760.

I am asking for examples of longer ordo sequences (see the records below).

Best regards,

Thomas Ordowski
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Records of length:
Initial odd term, length
3 15
563 21
1805 26
2131 38
5171 46
6207 47
11483 50
11909 52
34449 56
46997 73
70925 76
______________________
Records of last (even) term:
Initial odd term, last term
3 760
83 2536
411 4726
563 120004
1805 228286
2131 4505266
5171 6625126
11483 87570946
34449 131731066
46997 243223696
59441 649377226
134621 1115036206
____________________________
In both, the next term is > 300000.

Amiram Eldar