cp's OEIS Frontend

A096551 Consecutive internal states of a linear congruential pseudo-random number generator with a parameter proposed by George Marsaglia as a "candidate for the best of all multipliers".

%I A096551 #33 Jun 23 2025 19:35:47
%S A096551 1,69069,475559465,2801775573,1790562961,3104832285,4238970681,
%T A096551 2135332261,381957665,1744831853,1303896393,1945705589,2707602097,
%U A096551 4198202557,3820321881,201201733,2583294017,4003049741,2417848425,1454463253,3332335313,2360275549,2093206905,2813570789
%N A096551 Consecutive internal states of a linear congruential pseudo-random number generator with a parameter proposed by George Marsaglia as a "candidate for the best of all multipliers".
%C A096551 The period length of 2^30 is only achieved if the starting value a(1) is odd. Even starting values lead to shorter periods, i.e., a starting value that is a multiple of 2^k leads to a sequence with period length 2^(30-k). - _Hugo Pfoertner_, Nov 21 2024
%D A096551 D. E. Knuth, The Art of Computer Programming Third Edition. Vol. 2 Seminumerical Algorithms. Chapter 3.3.4 The Spectral Test, Page 108. Addison-Wesley 1997.
%D A096551 G. Marsaglia, The structure of linear congruential sequences, in Applications of Number Theory to Numerical Analysis, (edited by S. K. Zaremba), Academic Press, New York, 249-286, 1972.
%H A096551 Alois P. Heinz, <a href="/A096551/b096551.txt">Table of n, a(n) for n = 1..10000</a>
%H A096551 George S. Fishman, <a href="https://doi.org/10.2307/2008698">Multiplicative Congruential Random Number Generators with Modulus 2^beta: An Exhaustive Analysis for beta = 32 and a Partial Analysis for beta = 48</a>, Math. Comp., 54, 189 (1990), 331-344.
%H A096551 <a href="/index/Ps#PRN">Index entries for sequences related to pseudo-random numbers.</a>
%F A096551 a(1)=1, a(n) = 69069 * a(n-1) mod 2^32. The sequence is periodic with period length 2^30. - corrected by _Hugo Pfoertner_, Aug 10 2011
%F A096551 a(n) == 1 (mod 4). _Hugo Pfoertner_, Nov 21 2024
%p A096551 a:= proc(n) option remember; `if`(n<2, n,
%p A096551       irem(69069 *a(n-1), 4294967296))
%p A096551     end:
%p A096551 seq(a(n), n=1..30);  # _Alois P. Heinz_, Jun 10 2014
%t A096551 NestList[Mod[#*69069, 2^32] &, 1, 50] (* _Paolo Xausa_, Aug 29 2024 *)
%o A096551 (PARI) a(n)=lift(Mod(69069,2^32)^(n-1)) \\ _Charles R Greathouse IV_, Jan 14 2016
%Y A096551 Cf. A096550-A096561 (for other pseudo-random number generators).
%Y A096551 Cf. A385127 (same multiplier).
%K A096551 nonn,easy
%O A096551 1,2
%A A096551 _Hugo Pfoertner_, Jul 18 2004