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A326613
a(n) is the cyclic length of the iterative sequence f(k) = prime(f(k-1) mod 2^n) with f(0) = 1.
0
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).
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
Sequence in context: A085427 A172130 A202449 * A288627 A232096 A250030
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