[seqfan] Euler primary pretenders and periodicity of this sequence

[Tomasz Ordowski]

Dear SeqFans,

I have defined a periodic sequence that has a longer period than the
primary pretenders,
see https://oeis.org/A000790 and https://oeis.org/A108574 (the complete set
of terms).

Let a(n) be the smallest odd composite k such that n^{(k+1)/2} == +-n (mod
k), for n = 0,1,2,...

9, 9, 341, 121, 341, 15, 15, 21, 9, 9, 9, 33, 33, 21, 15, 15, 15, 9, 9, 9,
15, 15, 21, 33, 15, 15, 9, 9, 9, 15, 15, 15, 25, 33, 21, 9, 9, 9, 39, 15,
15, 21, 21, 21, 9, 9, 9, 65, 21, 21, 15, 15, 39, 9, 9, 9, 21, 21, 57, 15,
15, 15, 9, 9, 9, 15, 15, 33, 25, 15, 15, 9, 9, 9, 15, 15, 15, 21, 21, 39,
9, 9, 9, 21, 15, 15, 65, 33, 33, 9, 9, 9, 21, 25, 57, 15, 15, 21, 9, 9, 9,
...

The sequence is bounded, namely a(n) <= 1729, the smallest absolute Euler
pseudoprime,
because n^{(1729+1)/2} == n (mod 1729) for every n, see
https://oeis.org/A033181.

The set A = {a(n)} of terms contains all odd semiprimes pq < 1729 such that
p-1 | q-1.
The other numbers in this set are 561, 645, 1065, 1247, and 1729. Probably
complete.

A = {9, 15, 21, 25, 33, 39, 49, 51, 57, 65, 69, 85, 87, 91, 93, 111, 121,
123, 129, 133, 141, 145, 159, 169, 177, 183, 185, 201, 205, 213, 217, 219,
237, 249, 259, 265, 267, 289, 291, 301, 303, 305, 309, 321, 327, 339, 341,
361, 365, 381, 393, 411, 417, 427, 445, 447, 451, 453, 469, 471, 481, 485,
489, 501, 505, 511, 519, 529, 537, 543, 545, 553, 561, 565, 573, 579, 591,
597, 633, 645, 669, 671, 679, 681, 685, 687, 699, 703, 717, 721, 723, 745,
753, 763, 771, 781, 785, 789, 793, 807, 813, 831, 841, 843, 849, 865, 879,
889, 905, 921, 933, 939, 949, 951, 961, 965, 973, 985, 993, 1011, 1041,
1047, 1057, 1059, 1065, 1077, 1099, 1101, 1111, 1119, 1137, 1141, 1145,
1149, 1165, 1167, 1191, 1203, 1205, 1227, 1247, 1257, 1261, 1263, 1267,
1285, 1293, 1299, 1317, 1329, 1345, 1347, 1351, 1369, 1371, 1383, 1385,
1387, 1389, 1393, 1401, 1405, 1417, 1437, 1441, 1461, 1465, 1473, 1477,
1497, 1509, 1527, 1541, 1561, 1563, 1565, 1569, 1585, 1603, 1623, 1641,
1649, 1661, 1671, 1681, 1685, 1687, 1689, 1707, 1713, 1729}.

This sequence has the period P = LCM(A) = p# q# / 2^2,
where p and q are the largest primes such that p^2 < 1729 and 3q < 1729.
Such primes are p = 41 and q = 571, so P = 41# 571# / 4 (248 decimal
digits).

Are these known results, and this sequence has already been described in
the literature?

Thanks to Amiram Eldar and Daniel Suteu for help in numerical research of
this sequence.

Best regards,

Thomas Ordowski
________________________
Note that if p is an odd prime, then n^{(p+1)/2} == +-n (mod p) for all n.