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A144694 Initial terms from the linear congruential pseudorandom number generator defined by the recurrence a(n + 1) = a(n) * 7202161 + 1234567 (modulo 277945762500) with a(0) = 0.

%I A144694 #26 Jun 20 2025 17:06:48
%S A144694 0,1234567,275232896354,254512755561,100356676888,52443177035,
%T A144694 94402644702,182170448089,160360104896,160787264823,53709154570,
%U A144694 135639047837,234571597824,250315882231,144609285758,175142970105,222009343972,112485333059,70607625066,140050302193
%N A144694 Initial terms from the linear congruential pseudorandom number generator defined by the recurrence a(n + 1) = a(n) * 7202161 + 1234567 (modulo 277945762500) with a(0) = 0.
%C A144694 Referred to as ba3koxobe-cajedeja for convenience; this name is a simple representation of the modulus and multiplier in base 105.
%C A144694 The modulus 277945762500 is the fourth hyperprimorial, 2^2 * 3^3 * 5^5 * 7^7 (A076265).
%C A144694 The multiplier 7202161 is the semiprime 37 * 194653, located by exhaustive search of the integers allowing 64-bit computation. The maximum possible result of a(n) * 7202161 is 277945762499 * 7202161 = 2001810130785560339, a 61-bit value.
%C A144694 The addend 1234567 (A116935) is the memorable palindromic semiprime 127 * 9721, chosen for that property alone. Its value is not important, except that it must be nonzero and cannot be a multiple of 2, 3, 5, or 7.
%C A144694 Conversion of a(n) to 32 bits produces a sequence that passes the strongest empirical tests of randomness immediately available, including every test in the TestU01 Rabbit battery and every test in the Smallcrush battery except the birthday spacings test. Combining this output by XOR with the output of another generator, such as a Xorshift generator as described by Marsaglia, overcomes the observed flaws and allows the birthday spacings test to be passed as well.
%C A144694 Other multipliers identified for this modulus, believed to be marginally weaker (with minimal testing), in presumed order of decreasing strength, are 13683601, 19701361, 8692321, 11634841, 20627461, 3108421, 10419781, 3999241 and 9505021. All except 8692321 are semiprimes.
%C A144694 Reasons for the superiority of this modulus and these multipliers are guessed at, but remain to be determined.
%H A144694 Alois P. Heinz, <a href="/A144694/b144694.txt">Table of n, a(n) for n = 0..10000</a>
%H A144694 George Marsaglia, <a href="https://doi.org/10.18637/jss.v008.i14">Xorshift RNGs</a>, Journal of Statistical Software, Vol. 8, Issue 14, Jul 2003.
%H A144694 Kevin Ryde, <a href="/A144694/a144694.gp.txt">PARI/GP Code</a>
%H A144694 Richard Simard, <a href="https://simul.iro.umontreal.ca/testu01/tu01.html">TestU01: Empirical Testing of Random Number Generators</a>
%H A144694 John Walker, <a href="http://www.fourmilab.ch/random/">ENT: A Pseudorandom Number Sequence Test Program</a>
%H A144694 <a href="/index/Ps#PRN">Index entries for sequences related to pseudo-random numbers.</a>
%F A144694 a(n+1) = a(n) * 7202161 + c (modulo 277945762500) where c > 0 and gcd(c,277945762500) = 1, with a(0) a nonnegative integer.
%p A144694 a:= proc(n) option remember; `if`(n=0, 0,
%p A144694       irem(7202161 *a(n-1) +1234567, 277945762500))
%p A144694     end:
%p A144694 seq(a(n), n=0..30);  # _Alois P. Heinz_, Jun 10 2014
%t A144694 NestList[Mod[#*7202161 + 1234567, 277945762500] &, 0, 20] (* _Paolo Xausa_, Nov 13 2024 *)
%o A144694 (PARI) \\ See links.
%Y A144694 Cf. A076265, A116935.
%K A144694 easy,nonn,less
%O A144694 0,2
%A A144694 _Reikku Kulon_, Sep 19 2008