A144907 -id:A144907 - cp's OEIS frontend

A144908 Composite numbers n such that sqrt(n) > A144907(n) and absolute normalized digital mean |dm(b, n) * 2 / (b - 1)| decreases for b in [2, k] for some k > 2.

32, 64, 125, 128, 192, 250, 256, 288, 343, 375, 384, 500, 512, 576, 640, 648, 768, 800, 896, 1024, 1029, 1125, 1152, 1280, 1296, 1536, 1568, 1600, 1715, 1792, 1875, 2025, 2048, 2058, 2304, 2401, 2500, 2560, 2592, 2816, 3072, 3136, 3200, 3328, 3375, 3456

Offset: 1

Reikku Kulon, Sep 24 2008

Subset of A144100.

Believed to have particular importance for linear congruential pseudorandom number generators.

			125 is a member: A144907(125) is 5, which is less than 11.18, the square root of 125;
125 in base 2 is 1111101; dm(2, 125) = (6 * 1 - 1) / 14 = 5/14 ~ 0.357;
125 in base 3 is 11122; dm(3, 125) = (3 * 0 + 2 * 2) / 10 = 2/5 = 0.4;
125 in base 4 is 1331; dm(4, 125) = (2 * -1 + 2 * 3) / 8 = 1/2 = 0.5;
5/14 * 2 / 1 = 5/7 ~ 0.714;
2/5 * 2 / 2 = 2/5 = 0.4;
1/2 * 2 / 3 = 1/3 ~ 0.333;
For b in [2, 4], |dm(b, 125) * 2 / (b - 1)| is decreasing.

Cf. A144100, A144907, A144777

A144100 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.

Original entry on oeis.org

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, 47, 48, 49, 50, 53, 54, 56, 59, 61, 63, 64, 67, 71, 72, 73, 75, 79, 80, 81, 83, 88, 89, 90, 96, 97, 98, 99, 100, 101, 103, 104, 107, 108, 109, 112, 113, 117, 120, 121, 125, 126

Offset: 1

Reikku Kulon, Sep 10 2008

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.
4 is the only prime power not a member of the set: f(4) = 2 * rad(4) = 4.
This sequence consists of the primes and 2*A013929. - Charlie Neder, Jan 28 2019

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

T. D. Noe, Table of n, a(n) for n = 1..1000

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

a144100 n = a144100_list !! (n-1)
a144100_list = filter (\x -> a144907 x < x) [1..]
-- Reinhard Zumkeller, Mar 12 2014

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

A144907(a(n)) < a(n). - Reinhard Zumkeller, Mar 12 2014

cp's OEIS Frontend

A144908 Composite numbers n such that sqrt(n) > A144907(n) and absolute normalized digital mean |dm(b, n) * 2 / (b - 1)| decreases for b in [2, k] for some k > 2.

Views

Author

Keywords

Comments

Examples

Crossrefs

A144100 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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

PARI

Formula