A326613 a(n) is the cyclic length of the iterative sequence f(k) = prime(f(k-1) mod 2^n) with f(0) = 1.

3, 2, 1, 1, 2, 2, 5, 12, 14, 10, 33, 39, 174, 16, 95, 39, 413, 619, 949, 514, 2221, 3842, 2676, 2356, 5588, 8580, 2437, 7853, 14337, 26004, 1282, 72089, 42375, 24609, 82176, 124121, 289450, 713933, 321445, 730784, 3073601

Offset

2,1

Comments

Each iterative sequence is cyclic since there is a finite number of primes less than 2^k. Therefore a(n) <= A007053(n).

Links

Table of n, a(n) for n=2..42.

Example

For n=2, the seq. is 1, 2, 3, 5, 2, 3, 5, 2, 3, 5, 2, 3, 5, ..., ; which cycles every 3 terms;
for n=3, the seq. is 1, 2, 3, 5, 11, 5, 11, 5, 11, 5, 11, 5, ... ; which cycles every 2 terms;
for n=4, the seq. is 1, 2, 3, 5, 11, 31, 47, 47, 47, ..., ; which cycles every term;
for n=5, the seq. is 1, 2, 3, 5, 11, 31, 127, 127, 127, ..., ; which cycles every term;
for n=6, the seq. is 1, 2, 3, 5, 11, 31, 127, 307, 233, 179, 233, 179, 233, ..., ; which cycles every 2 terms;
for n=7, the seq. is 1, 2, 3, 5, 11, 31, 127, 709, 347, 467, 431, 211, 431, 211, 431, ..., ; which cycles every 2 terms;
for n=8, the seq. is 1, 2, 3, 5, 11, 31, 127, 709, 1201, 1051, 103, 563, 233, 1471, 1153, 727, 1319, 167, 991, 1409, 727, 1319, 167, 991, 1409, ..., ; which cycles every 5 terms;
for n=9, the seq. is 1, 2, 3, 5, 11, 31, 127, 709, 1201, 1051, 103, 563, 233, 1471, 3163, 467, 3319, 1567, 127, 709, 1201, 1051, 103, 563, 233, 1471, 3163, 467, 3319, 1567, 127, ..., ; which cycles every 12 terms;
for n=10, the seq. is 1, 2, 3, 5, 11, 31, 127, 709, 5381, 1663, 4733, 4723, 4643, 3943, 6763, 4567, 3343, 1741, 5431, 2063, 47, 211, 1297, 1753, 5519, 2731, 5107, 8039, 6763, 4567, 3343, 1741, 5431, 2063, 47, 211, 1297, 1753, 5519, 2731, 5107, 8039, 6763, ..., ; which cycles every 14 terms;
for n=11, the seq. is 1, 2, 3, 5, 11, 31, 127, 709, 5381, 10501, 1663, 14107, 15601, 10313, 367, 2477, 2971, 7219, 8629, 3049, 7927, 15271, 7333, 9629, 11981, 14867, 3823, 15199, 6691, 3943, 16339, 17417, 8233, 179, 1063, 8527, 2251, 1237, 10079, 16231, 16339, 17417, 8233, 179, 1063, 8527, 2251, 1237, 10079, 16231, 16339, 17417, 8233, 179, 1063, 8527, 2251, 1237, 10079, 16231, 16339, 17417, 8233, 179, 1063, 8527, 2251, 1237, 10079, 16231, 16339; which cycles every 10 terms;
... as n-> inf. the seq. is 1, 2, 3, 5, 11, 31, 127, 709, 5381, 52711, 648391, ..., .

Mathematica

f[n_] := Block[{s = NestWhileList[ Prime[ Mod[#, 2^n]] &, 1, UnsameQ, All]}, s = -Subtract @@ Position[s, s[[-1]]]; s[[1]]]

Crossrefs

Cf. A007097, A128867.

Sequence in context: A085427 A172130 A202449 * A288627 A232096 A250030

Adjacent sequences: A326610 A326611 A326612 * A326614 A326615 A326616

Keyword

nonn,more

Author

Robert G. Wilson v, Sep 13 2019

Extensions

a(35) corrected and a(37)-a(41) added by Chai Wah Wu, Oct 02 2019

a(42) from Chai Wah Wu, Oct 07 2019

Status

approved