%I #29 Jan 04 2022 21:46:03

%S 3,5,7,17,11,13,31,41,19,53,23,97,59,29,47,89,163,37,103,113,43,109,

%T 127,73,179,149,311,193,283,61,79,313,67,197,71,337,331,461,223,233,

%U 83,373,599,673,251,349,239,281,227,101,167,433,107,173,191,137,499,853,503,257,347,509,271,761,131,389,647,401,139,397,1439,937,659,181,151,1249,443,157,431,1753,547,421,487,241,683,2029,1279,457,307,277,439,577,2203,701,463,953,1987,1093,199,977

%N A permutation of the odd primes that satisfy the condition that the sequence modulo 2^n has period 2^(n-1) for all n>0, where the least unused primes are chosen in the process.

%C Index of primes is given by A099864.

%H Paul D. Hanna, <a href="/A099863/b099863.txt">Table of n, a(n) for n = 1..32768</a>

%e The sequence is of period 2^(n-1) modulo 2^n, for all n>0 and consists of all odd numbers less than 2^n:

%e A (mod 2) = [1, ... (repeating)];

%e A (mod 4) = [3, 1, ... (repeating)];

%e A (mod 8) = [3, 5, 7, 1, ... (repeating)];

%e A (mod 16) = [3, 5, 7, 1, 11, 13, 15, 9, ... (repeating)];

%e A (mod 32) = [3, 5, 7, 17, 11, 13, 31, 9, 19, 21, 23, 1, 27, 29, 15, 25, ... (repeating)];

%e A (mod 64) = [3, 5, 7, 17, 11, 13, 31, 41, 19, 53, 23, 33, 59, 29, 47, 25, 35, 37, 39, 49, 43, 45, 63, 9, 51, 21, 55, 1, 27, 61, 15, 57, ... (repeating)];

%e A (mod 128) = [3, 5, 7, 17, 11, 13, 31, 41, 19, 53, 23, 97, 59, 29, 47, 89, 35, 37, 103, 113, 43, 109, 127, 73, 51, 21, 55, 65, 27, 61, 79, 57, 67, 69, 71, 81, 75, 77, 95, 105, 83, 117, 87, 33, 123, 93, 111, 25, 99, 101, 39, 49, 107, 45, 63, 9, 115, 85, 119, 1, 91, 125, 15, 121, ... (repeating)];

%e A (mod 256) = [3, 5, 7, 17, 11, 13, 31, 41, 19, 53, 23, 97, 59, 29, 47, 89, 163, 37, 103, 113, 43, 109, 127, 73, 179, 149, 55, 193, 27, 61, 79, 57, 67, 197, 71, 81, 75, 205, 223, 233, 83, 117, 87, 161, 251, 93, 239, 25, 227, 101, 167, 177, 107, 173, 191, 137, 243, 85, 247, 1, 91, 253, 15, 249, 131, 133, 135, 145, 139, 141, 159, 169, 147, 181, 151, 225, 187, 157, 175, 217, 35, 165, 231, 241, 171, 237, 255, 201, 51, 21, 183, 65, 155, 189, 207, 185, 195, 69, 199, 209, 203, 77, 95, 105, 211, 245, 215, 33, 123, 221, 111, 153, 99, 229, 39, 49, 235, 45, 63, 9, 115, 213, 119, 129, 219, 125, 143, 121, ... (repeating)];

%e A (mod 512) = [3, 5, 7, 17, 11, 13, 31, 41, 19, 53, 23, 97, 59, 29, 47, 89, 163, 37, 103, 113, 43, 109, 127, 73, 179, 149, 311, 193, 283, 61, 79, 313, 67, 197, 71, 337, 331, 461, 223, 233, 83, 373, 87, 161, 251, 349, 239, 281, 227, 101, 167, 433, 107, 173, 191, 137, 499, 341, 503, 257, 347, 509, 271, 249, 131, 389, 135, 401, 139, 397, 415, 425, 147, 181, 151, 225, 443, 157, 431, 217, 35, 421, 487, 241, 171, 493, 255, 457, 307, 277, 439, 65, 155, 189, 463, 441, 451, 69, 199, 465, 459, 77, 95, 105, 211, 245, 215, 33, 379, 221, 367, 409, 99, 229, 39, 49, 491, 45, 63, 9, 371, 213, 119, 129, 475, 125, 399, 121, 259, 261, 263, 273, 267, 269, 287, 297, 275, 309, 279, 353, 315, 285, 303, 345, 419, 293, 359, 369, 299, 365, 383, 329, 435, 405, 55, 449, 27, 317, 335, 57, 323, 453, 327, 81, 75, 205, 479, 489, 339, 117, 343, 417, 507, 93, 495, 25, 483, 357, 423, 177, 363, 429, 447, 393, 243, 85, 247, 1, 91, 253, 15, 505, 387, 133, 391, 145, 395, 141, 159, 169, 403, 437, 407, 481, 187, 413, 175, 473, 291, 165, 231, 497, 427, 237, 511, 201, 51, 21, 183, 321, 411, 445, 207, 185, 195, 325, 455, 209, 203, 333, 351, 361, 467, 501, 471, 289, 123, 477, 111, 153, 355, 485, 295, 305, 235, 301, 319, 265, 115, 469, 375, 385, 219, 381, 143, 377, ... (repeating)];

%e ...

%e The last prime in the first 2^(n-1) terms of A (mod 2^n), n >= 2, begins

%e [3, 7, 13, 29, 61, 107, 229, 467, 991, 1667, 3271, 8147, 16339, 32303, ...].

%e Position of the last prime in the first 2^(n-1) terms of A (mod 2^n), n >= 2:

%e [1, 3, 6, 14, 30, 53, 114, 233, 423, 833, 1635, 3561, 7657, 15631, ...].

%e The position of the first occurrence of 1 in A (mod 2^n), n >= 1, begins:

%e [1, 2, 4, 4, 12, 28, 60, 60, 188, 444, 444, 444, 2492, 6588, 6588, ...].

%e The position of the first occurrence of 2^n-1 in A (mod 2^n), n >= 1, begins:

%e [1, 1, 3, 7, 7, 23, 23, 87, 215, 471, 983, 2007, 2007, 6103, 14295, ...].

%e The value of a(2^n) (mod 2^n), n >= 1, begins

%e [1, 1, 1, 9, 25, 57, 121, 121, 377, 377, 377, 377, 377, 377, 377, ...].

%e The value of a(2^n), n >= 0, begins

%e [3, 5, 17, 41, 89, 313, 761, 1657, 1913, 3449, 6521, 20857, 24953, 49529, 131449, 229753, ...].

%e Positions of prime(n), for n >= 2, begin

%e [1, 2, 3, 5, 6, 4, 9, 11, 14, 7, 18, 8, 21, 15, 10, 13, 30, 33, 35, 24, 31, 41, 16, 12, 50, 19, 53, 22, 20, 23, 65, 56, 69, 26, 75, 78, 17, 51, 54, 25, 74, 55, 28, 34, 99, 105, 39, 49, 114, 40, 47, 84, 45, 60, 131, 134, 63, 90, 48, 29, 146, 89, 27, 32, 158, 37, 36, 61, 46, 140, 147, 111, 42, 109, 151, 66, 70, 68, 112, 145, 82, 79, 52, 91, 77, 156, 88, 38, 95, 233, 167, 83, 117, 57, 59, 62, 120, 261, 270, 81, 118, 217, 160, 269, 92, 165, 164, 43, 72, 103, 306, 104, 309, ...].

%e RECORDS.

%e Records in this sequence begin:

%e [3, 5, 7, 17, 31, 41, 53, 97, 163, 179, 311, 313, 337, 461, 599, 673, 853, 1439, 1753, 2029, 2203, 2609, 3709, 4663, 7681, 7993, 16963, 20921, 25603, 36523, 38803, 38959, 59471, 75913, 82141, 87959, 106417, 110609, 135241, 160207, 171733, 171799, 265547, 321647, 630893, 638663, 704861, 836833, 1002083, 1067653, 1440107, 1634201, 2588983, 2861569, 4668899, ...].

%e Positions of records begin:

%e [1, 2, 3, 4, 7, 8, 10, 12, 17, 25, 27, 32, 36, 38, 43, 44, 58, 71, 80, 86, 93, 116, 126, 155, 188, 288, 289, 352, 513, 597, 969, 1039, 1055, 1080, 1390, 1739, 1844, 2052, 2072, 2143, 2426, 2443, 2597, 3055, 4118, 4451, 8238, 8268, 8561, 8898, 10421, 15216, 16411, 17084, 18097, ...].

%o (PARI) /* Print the first 2^L terms (size M of P vector may need adjustment): */

%o {L=10; M=4*L*2^L; A=vector(2^L); P=vector(M); A[1]=3; P[1]=1;

%o for(i=1,L, for(n=2^(i-1)+1,2^i, for(m=1,M,q=A[n-2^(i-1)]+(2*m-1)*2^i; if(isprime(q)&P[q]==0, A[n]=q; P[q]=1; next(2)) )));A}

%Y Cf. A099864.

%K nonn

%O 1,1

%A _Paul D. Hanna_, Oct 27 2004

%E Added more terms, expanded b-file, and expanded Example section. - _Paul D. Hanna_, Aug 18 2019