[seqfan] Re: The ord-sequences

[Tomasz Ordowski]

P.S. I noticed that the ord-sequences can be extended to infinity by the
recursion:

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

Formula: a(n+1) = a(n) + ord_{Od(a(n))}(2), where Od(m) is the odd part of
m.

For a(1) = 1, we get the sequence:
1, 2, 3, 5, 9, 15, 19, 37, 73, 82, 102, 110, 130, 142, 177, 235, 327, 363,
473, 543, 723, 747, 993, 1023, 1033, 1291, 2581, 2889, 3843, 3903, 5203,
5973, 6153, 7029, 7239, 7365, 8345, 10013, 10373, 10593, 12183, 12313,
13192, 13240, 13300, 13480, 13564, 13677, ...

The number of consecutive terms with the same parity:
1,1,7,5,28,5,16,13,47,4,70,6,56,4,32,11,17,21,22,11,2,20,67,13,22,36,8,9,101,47,24,4,1,7,2,79,20,71,47,92,36,57,90,38,167,215,31,17,62,...

i.e.,1 odd term, 1 even term, 7 odd terms, 5 even terms, 28 odd terms, 5
even terms, ...


niedz., 3 mar 2019 o 16:18 Tomasz Ordowski <tomaszordowski at gmail.com>
napisał(a):

> Dear SeqFans!
>
> Consider the ord-sequences defined by the following recursion:
> a(n+1) is the smallest k > a(n) such that 2^{k-a(n)} == 1 (mod a(n)).
> For example: 3, 5, 9, 15, 19, 37, 73, and 82. Ended with an even number.
> Conjecture: For any initial term a(1), ord-sequence is finite (an even
> term).
> Note that a(n+1) = a(n) + ord_{a(n)}(2). Hence the name: ord-sequences.
> Theorem: a(n+1) is even if and only if ord_{a(n)}(2) is odd. See A036259.
> Question: Will each ord-sequence hit such a penultimate number?
> I'm asking for new length records of my sequences (see below).
>
> Best regards,
>
> Thomas Ordowski
> _______________
> The length records:
> a(1), length, last term;
> 1, 2, 2
> 3, 8, 82
> 11, 18, 2246
> 39, 33, 13192
> 179, 54, 811840
> 197, 83, 140596630
> 2481, 115, 1239299494
> 6299, 191, 263364616132
> 36365, 269, 240155184043288
> 231393, 312, 648274634463250
> 249971, 358, 124677735534497570
> 2656943, 394, ??????????????????
> 5006999, 439, 15748956693023917552
> 5745479, 477, 560480869962021796634
> 58182659, 523, 1051892735063463075706
> [Amiram Eldar]
>
>