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Carmichael numbers ( A002997) that are Chebyshev pseudoprimes ( A175530).
+20
2
443372888629441, 582920080863121, 39671149333495681, 842526563598720001, 2380296518909971201, 3188618003602886401, 33711266676317630401, 54764632857801026161, 122762671289519184001, 361266866679292635601, 4208895375600667752001, 7673096805497432749441
COMMENTS
Odd composite integer n is in this sequence if n == 1 or p (mod (p^2-1)/2) for every prime p|n.
No other terms below 10^22.
Chebyshev pseudoprimes to base 2: composite numbers k such that T(k, 2) == 2 (mod k), where T(k, x) is the k-th Chebyshev polynomial of the first kind.
+10
4
209, 231, 399, 455, 901, 903, 923, 989, 1295, 1729, 1855, 2015, 2211, 2345, 2639, 2701, 2795, 2911, 3007, 3201, 3439, 3535, 3801, 4823, 5291, 5719, 6061, 6767, 6989, 7421, 8569, 9503, 9591, 9869, 10439, 10609, 11041, 11395, 11951, 11991, 13133, 13529, 13735, 13871
COMMENTS
Bang proved that T(p, a) == a (mod p) for every a > 0 and every odd prime. Rayes et al. (1999) defined Chebyshev pseudoprimes to base a as composite numbers k such that T(k, a) == a (mod k). They noted that the first Chebyshev pseudoprime to base 2 is 209.
EXAMPLE
209 is in the sequence since 209 = 11 * 19 is composite and T(209, 2) - 2 is divisible by 209.
MATHEMATICA
Select[Range[15000], CompositeQ[#] && Divisible[ChebyshevT[#, 2] - 2, #] &]
Carmichael numbers of order 2.
+10
3
443372888629441, 39671149333495681, 842526563598720001, 2380296518909971201, 3188618003602886401, 4208895375600667752001
COMMENTS
Odd composite integer k is in this sequence if k == 1 or p (mod p^2 - 1) for every prime p|k.
Chebyshev pseudoprimes to base 3: composite numbers k such that T(k, 3) == 3 (mod k), where T(k, x) is the k-th Chebyshev polynomial of the first kind.
+10
3
14, 35, 119, 169, 385, 434, 574, 741, 779, 899, 935, 961, 1105, 1106, 1121, 1189, 1443, 1479, 2001, 2419, 2555, 2915, 3059, 3107, 3383, 3605, 3689, 3741, 3781, 3827, 4199, 4795, 4879, 4901, 5719, 6061, 6083, 6215, 6265, 6441, 6479, 6601, 6895, 6929, 6931, 6965
COMMENTS
Bang proved that T(p, a) == a (mod p) for every a > 0 and every odd prime. Rayes et al. (1999) defined Chebyshev pseudoprimes to base a as composite numbers k such that T(k, a) == a (mod k).
EXAMPLE
14 is in the sequence since it is composite and T(14, 3) = 26102926097 == 3 (mod 14).
MATHEMATICA
Select[Range[1000], CompositeQ[#] && Divisible[ChebyshevT[#, 3] - 3, #] &]
Chebyshev pseudoprimes to both base 2 and base 3: composite numbers k such that T(k, 2) == 2 (mod k) and T(k, 3) == 3 (mod k), where T(k, x) is the k-th Chebyshev polynomial of the first kind.
+10
3
5719, 6061, 11395, 15841, 17119, 18721, 31535, 67199, 73555, 84419, 117215, 133399, 133951, 174021, 181259, 194833, 226801, 273239, 362881, 469201, 516559, 522899, 534061, 588455, 665281, 700321, 721801, 778261, 903959, 1162349, 1561439, 1708901, 1755001, 1809697
COMMENTS
Bang proved that T(p, a) == a (mod p) for every a > 0 and every odd prime. Rayes et al. (1999) defined Chebyshev pseudoprimes to base a as composite numbers k such that T(k, a) == a (mod k). They noted that there are no Chebyshev pseudoprimes in both bases 2 and 3 below 2000.
EXAMPLE
5719 is in the sequence since 5719 = 7 * 19 * 43 is composite and both T(5719 , 2) - 2 and T(5719, 3) - 3 are divisible by 5719.
MATHEMATICA
Select[Range[2*10^4], CompositeQ[#] && Divisible[ChebyshevT[#, 2] - 2, #] && Divisible[ChebyshevT[#, 3] - 3, #] &]
Squarefree composite numbers n such that p^2 - 1 divides n^2 - 1 for every prime p dividing n.
+10
1
8569, 39689, 321265, 430199, 564719, 585311, 608399, 7056721, 11255201, 17966519, 18996769, 74775791, 75669551, 136209151, 321239359, 446660929, 547674049, 866223359, 1068433631, 1227804929, 1291695119, 2315403649, 2585930689, 7229159729, 7809974369, 8117634239
COMMENTS
Such numbers are odd and have at least three prime factors.
Problem: are there infinitely many such numbers?
PROG
(PARI) isok(n) = {if (issquarefree(n) && !isprime(n), my(f = factor(n)); for (k=1, #f~, if ((n^2-1) % (f[k, 1]^2-1), return (0)); ); return (1); ); } \\ Michel Marcus, May 20 2017
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