A144100 - OEIS

%I #19 Jul 16 2021 03:25:33

%S 2,3,5,7,8,9,11,13,16,17,18,19,23,24,25,27,29,31,32,36,37,40,41,43,45,

%T 47,48,49,50,53,54,56,59,61,63,64,67,71,72,73,75,79,80,81,83,88,89,90,

%U 96,97,98,99,100,101,103,104,107,108,109,112,113,117,120,121,125,126

%N Numbers k such that k is strictly greater than f(k), where f(k) = 1 if k is prime, 2 * rad(k) if 4 divides k and rad(k) otherwise.

%C This is the set of all integers k such that there exists a full period linear congruential pseudorandom number generator x -> bx + c (mod k), where b is not a multiple of k, b - 1 is a multiple of f(k) and c is a positive integer relatively prime to k.

%C 4 is the only prime power not a member of the set: f(4) = 2 * rad(4) = 4.

%C This sequence consists of the primes and 2*A013929. - _Charlie Neder_, Jan 28 2019

%H T. D. Noe, <a href="/A144100/b144100.txt">Table of n, a(n) for n = 1..1000</a>

%F A144907(a(n)) < a(n). - _Reinhard Zumkeller_, Mar 12 2014

%e 2 is a member: f(2) = 1 and the sequence (0, 1, 0, ...) given by x -> x + 1 (mod 2) has period 2.

%e 8 is a member: f(8) = 4 and the sequence (0, 1, 6, 7, 4, 5, 2, 3, 0, ...) given by x -> 5x + 1 (mod 8) has period 8.

%e 18 is a member: f(18) = 6 and the sequence (0, 1, 14, 3, 4, 17, 6, 7, 2, 9, 10, 5, 12, 13, 8, 15, 16, 11, 0, ...) given by x -> 13x + 1 (mod 18) has period 18.

%o (PARI) rad(n) = local(p); p=factor(n)[, 1]; prod(i=1, length(p), p[i]) ;

%o f(n) = if (isprime(n), 1, if ((n % 4)==0 , 2*rad(n), rad(n))); isok(n) = n > f(n); \\ _Michel Marcus_, Aug 09 2013

%o (Haskell)

%o a144100 n = a144100_list !! (n-1)

%o a144100_list = filter (\x -> a144907 x < x) [1..]

%o -- _Reinhard Zumkeller_, Mar 12 2014

%Y Cf. A000040, A007947, A071139, A082377, A086486, A089352, A133810, A133811.

%K easy,nice,nonn

%O 1,1

%A _Reikku Kulon_, Sep 10 2008