A096561 -id:A096561 - cp's OEIS frontend

A096550 Consecutive internal states of the IMSL pseudo-random number generator RNUN when started with ISEED=1.

1, 16807, 282475249, 1622650073, 984943658, 1144108930, 470211272, 101027544, 1457850878, 1458777923, 2007237709, 823564440, 1115438165, 1784484492, 74243042, 114807987, 1137522503, 1441282327, 16531729, 823378840, 143542612, 896544303, 1474833169, 1264817709, 1998097157

Offset: 1

Hugo Pfoertner, Jul 18 2004

This generator is also called "The minimal standard generator" or LCG16807 by L'Ecuyer. Generators of this form are ascribed to D. H. Lehmer, first described by Hutchinson and independently by Downham and Roberts (see link). It was first analyzed by Lewis, Goodman and Miller (see link).

Also used by Lotus 1-2-3 and some versions of APL. - Sean A. Irvine, May 27 2025

D. W. Hutchinson, A new uniform pseudo-random number generator. Comm, ACM 9, No. 6, 432-433, 1966.
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.

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from Eric M. Schmidt)
D. Y. Downham and F. D. K. Roberts, Multiplicative congruential pseudo-random number generators. The Computer Journal, Volume 10, Issue 1, 74-77.
George S. Fishman and Louis R. Moore, A Statistical Evaluation of Multiplicative Congruential Random Number Generators with Modulus 2^31-1, J. American Statistical Assoc., 77, 377 (1982), 129-136.
Pierre L'Ecuyer, Efficient and portable combined random number generators, C ACM, 31, 6 (1988), 742-749 and 774.
Pierre L'Ecuyer, Software for Uniform Random Number Generation: Distinguishing the Good and the Bad. Proceedings of the 2001 Winter Simulation Conference, IEEE Press, Dec. 2001, 95-105.
P. A. W. Lewis, A. S. Goodman and J. M. Miller, A pseudo-random number generator for the System/360, IBM Systems Journal, Volume 8 Issue 2, 136-146, 1969.
Stephen K. Park and Keith. W. Miller, Random Number Generators: Good Ones are Hard to Find, Communications of the ACM, Volume 31, Number 10 (October, 1988), 1192-1201.
B. D. Ripley, Computer Generation of Random Variables: A Tutorial, International Statistical Review, 51 (1983), 301-309.
B. D. Ripley, Thoughts on pseudorandom number generators, J of Computational and Applied Mathematics, 31, 1 (1990), 153-163.
Index entries for sequences related to pseudo-random numbers.

Cf. A096551-A096561 (other pseudo-random number generators); A061364, A384406.

a:= proc(n) option remember; `if`(n<2, n,
      irem(16807 *a(n-1), 2147483647))
    end:
seq(a(n), n=1..30);  # Alois P. Heinz, Jun 10 2014

NestList[Mod[#*16807, 2^31 - 1] &, 1, 50] (* Paolo Xausa, Aug 29 2024 *)

A096550(n)=lift(Mod(16807,1<<31-1)^(n-1)) \\ M. F. Hasler, May 14 2015

a(1)=1, a(n) = 7^5 * a(n-1) mod (2^31-1). The sequence is periodic with period length 2^31-2.

A384113 Consecutive states of a linear congruential pseudo-random number generator for MacModula-2 when started at 1.

Original entry on oeis.org

1, 13, 169, 2197, 829, 1533, 1441, 245, 874, 2118, 2113, 2048, 1203, 1773, 2250, 1518, 1246, 21, 273, 1238, 2228, 1232, 2150, 218, 523, 2177, 569, 464, 1410, 2153, 257, 1030, 1835, 745, 441, 1111, 577, 568, 451, 1241, 2267, 1739, 1808, 394, 500, 1878, 1304

Offset: 1

Sean A. Irvine, May 19 2025

An example of a terrible random number generator.

Periodic with period 1155 (well below the modulus 2311).

Modula Corporation, MacModula-2 System Reference Manual, 1985 (see p. 41).

Sean A. Irvine, Table of n, a(n) for n = 1..1155
Stephen K. Park and Keith W. Miller, Random number generators: good ones are hard to find, Communications of the ACM, Vol 31, 10 (1988), 192-201.
Index entries for sequences related to pseudo-random numbers.
Index entries for linear recurrences with constant coefficients, order 1155.

Cf. A001022.
Cf. A096550-A096561 other pseudo-random number generators.
Cf. A383809 (another generator with a similar problem).

a:= proc(n) option remember; `if`(n<2, n,
      irem(13*a(n-1), 2311))
    end:
seq(a(n), n=1..47);  # Alois P. Heinz, May 21 2025

NestList[Mod[13*#, 2311] &, 1, 100] (* Paolo Xausa, May 22 2025 *)

my(f=Mod(13,2311)); a(n) = lift(f^((n-1) % 1155)); \\ Kevin Ryde, May 25 2025

a(n) = 13 * a(n-1) mod 2311.

A384489 Consecutive states of the linear congruential pseudo-random number generator 392314069 * s mod 2^32 when started at s=1.

Original entry on oeis.org

1, 392314069, 3884484921, 1268090989, 4095610545, 2939532613, 4120247913, 1352616285, 3662927457, 371333813, 3840713881, 2970275661, 487491345, 3493879077, 1452026825, 2933230141, 3932967105, 2951638165, 920470521, 3864652333, 1810654065, 1799305477

Offset: 1

Sean A. Irvine, May 30 2025

Periodic with period 2^30.

George S. Fishman, Discrete-Event Simulation: Modeling, Programming, and Analysis, Springer, 2001 (see p. 453).

Sean A. Irvine, Table of n, a(n) for n = 1..10000
George S. Fishman, Multiplicative Congruential Random Number Generators with Modulus 2^beta: An Exhaustive Analysis for beta = 32 and a Partial Analysis for beta = 48, Math. Comp., 54, 189 (1990), 331-344.
Index entries for sequences related to pseudo-random numbers.

Cf. A384534.

Cf. A096550-A096561 other pseudo-random number generators.

NestList[Mod[392314069*#, 2^32] &, 1, 50] (* Paolo Xausa, Jun 04 2025 *)

a(n) = 392314069 * a(n-1) mod 2^32.

A384158 Consecutive states of the linear congruential pseudo-random number generator for 16-bit WATFOR/WATFIV when started at 1.

Original entry on oeis.org

1, 253, 31241, 6885, 5201, 5133, 20697, 26229, 16801, 23581, 2217, 3845, 22513, 26925, 29049, 9365, 10049, 19261, 23369, 14117, 32657, 4685, 5657, 22197, 12513, 20061, 29161, 4933, 2865, 3949, 16057, 31957, 24193, 25981, 19593, 9061, 31441, 24717, 27481, 5877

Offset: 1

Sean A. Irvine, May 20 2025

Periodic with period 8192 (considerably less than the modulus).

WATFOR and WATFIV are early FORTRAN compilers from the University of Waterloo.

Terry M. Walker, Fundamentals of Fortran Programming: with WATFOR/WATFIV, Allyn and Bacon, 1975.

Sean A. Irvine, Table of n, a(n) for n = 1..8192
Index entries for sequences related to pseudo-random numbers.
Index entries for linear recurrences with constant coefficients, order 8192.

Cf. A384159, A384160.

Cf. A096550-A096561 other pseudo-random number generators.

a:= proc(n) option remember; `if`(n<2, n,
      irem(253*a(n-1), 2^15))
    end:
seq(a(n), n=1..40);  # Alois P. Heinz, May 21 2025

NestList[Mod[253*#, 2^15] &, 1, 100] (* Paolo Xausa, May 22 2025 *)

a(n) = 253 * a(n-1) mod 2^15.

a(n) == 1 (mod 4). - Hugo Pfoertner, May 26 2025

A384081 Consecutive internal states of a linear congruential pseudo-random number generator for the Hewlett-Packard HP-20S when started at 1.

Original entry on oeis.org

1, 997, 994009, 1026973, 3892081, 404757, 3542729, 2100813, 4510561, 7029317, 8229049, 4361853, 8767441, 1138677, 5260969, 5186093, 534721, 3116837, 7486489, 4029533, 7444401, 2067797, 1593609, 8828173, 1688481, 3415557, 5310329, 4398013, 4818961, 4504117

Offset: 1

Sean A. Irvine, May 23 2025

Periodic with period 500000 (considerably less than the modulus 10^7).

Sean A. Irvine, Table of n, a(n) for n = 1..10000
Hewlett-Packard, HP-20S Scientific Calculator: Owner's Manual, 1992 (see p. 75).
W. E. Sharp and Carter Bays, A review of portable random number generators, Computers and Geosciences, 18, 1 (1982), 79-87.
Index entries for sequences related to pseudo-random numbers.
Index entries for linear recurrences with constant coefficients, order 500000.

Cf. A096550-A096561 (other pseudo-random number generators).

Cf. A384568.

NestList[Mod[997*#, 10^7] &, 1, 50] (* Paolo Xausa, May 26 2025 *)

a(n) = (997 * a(n-1)) mod 10^7.

A384126 Consecutive states of a linear congruential pseudo-random number generator (93*s+1) mod 2^13 when started at s=1.

Original entry on oeis.org

1, 94, 551, 2092, 6141, 5866, 4867, 2072, 4281, 4918, 6815, 3012, 1589, 322, 5371, 7984, 5233, 3342, 7703, 3676, 5997, 666, 4595, 1352, 2857, 3558, 3215, 4084, 2981, 6898, 2539, 6752, 5345, 5566, 1543, 4236, 733, 2634, 7395, 7800, 4505, 1174, 2687, 4132, 7445

Offset: 1

Sean A. Irvine, May 19 2025

Periodic with period 8192.

John Konvalina and Stanley Wileman, Programming with Pascal, McGraw-Hill, 1987 (see p. 288).

Sean A. Irvine, Table of n, a(n) for n = 1..8192
Stephen K. Park and Keith W. Miller, Random number generators: good ones are hard to find, Communications of the ACM, Vol 31, 10 (1988), 192-201.
W. E. Sharp and Carter Bays, A review of portable random number generators, Computers and Geosciences, 18, 1 (1982), 79-87.
Index entries for sequences related to pseudo-random numbers.
Index entries for linear recurrences with constant coefficients, order 8192.

Cf. A096550-A096561 (other pseudo-random number generators).

a:= proc(n) option remember; `if`(n<2, n,
      irem(93*a(n-1)+1, 2^13))
    end:
seq(a(n), n=1..45);  # Alois P. Heinz, May 21 2025

NestList[Mod[93*# + 1, 2^13] &, 1, 100] (* Paolo Xausa, May 22 2025 *)

a(n) = (93*a(n-1) + 1) mod 2^13.

A384289 Consecutive internal states of the linear congruential pseudo-random number generator for GWBASIC 3.23 when started at 1.

Original entry on oeis.org

1, 2745024, 2356867, 12486458, 8679701, 14802820, 7082039, 14027294, 11434089, 5380488, 9466411, 4830274, 15796733, 15840460, 12300383, 15321510, 15423953, 11736400, 10919635, 14405194, 3988453, 8904468, 807303, 4097582, 10044473, 2422296, 6167675, 914770

Offset: 1

Sean A. Irvine, May 24 2025

Periodic with period 2^24.
Also the random number generator used by Commodore AmigaBASIC.
Possibly also used by other versions of GWBASIC.

David I. Schneider, Handbook of BASIC (3rd ed.), Simon & Schuster, 1988 (see p. 497).

Sean A. Irvine, Table of n, a(n) for n = 1..10000
W. E. Sharp and Carter Bays, A review of portable random number generators, Computers and Geosciences, 18, 1 (1982), 79-87.
Index entries for sequences related to pseudo-random numbers.

Cf. A096550-A096561 other pseudo-random number generators.

Cf. A384290, A384291 (similar parameters).

NestList[Mod[214013*#+2531011, 2^24] &, 1, 27] (* Stefano Spezia, May 24 2025 *)

a(n) = (214013 * a(n-1) + 2531011) mod 2^24.

A384397 Consecutive states of the linear congruential pseudo-random number generator 45742*s mod (2^31-909) when started at s=1.

Original entry on oeis.org

1, 45742, 2092330564, 521429475, 1283746116, 346822856, 916086159, 1929881610, 71652547, 472145160, 1777485336, 2037740772, 997589268, 2015058584, 703108709, 897067814, 1723253915, 1726644935, 72441828, 68230099, 688768691, 2085682592, 1372443189, 933442051

Offset: 1

Sean A. Irvine, May 27 2025

Periodic with period 2^31-910.

Sean A. Irvine, Table of n, a(n) for n = 1..10000
Pierre L'Ecuyer, Efficient and portable combined random number generators, Commun. ACM, 31, 6 (1988), 742-749 and 774.
Index entries for sequences related to pseudo-random numbers.

Cf. A384398, A384399, A384400, A384401, A384402.

Cf. A096550-A096561 other pseudo-random number generators.

NestList[Mod[45742*#, 2^31 - 909] &, 1, 50] (* Paolo Xausa, May 29 2025 *)

a(n) = 45742 * a(n-1) mod (2^31-909).

A384398 Consecutive states of the linear congruential pseudo-random number generator 42024*s mod (2^31-847) when started at s=1.

Original entry on oeis.org

1, 42024, 1766016576, 222470065, 1089378807, 16633650, 1080597275, 348574654, 521074075, 1882288804, 923207262, 437695422, 522223563, 796268093, 293335050, 560863460, 1102302065, 1937961990, 1924405437, 1306764430, 38219148, 1951823205, 312782725, 1786493280

Offset: 1

Sean A. Irvine, May 27 2025

Periodic with period 2^31-848.

Sean A. Irvine, Table of n, a(n) for n = 1..10000
Pierre L'Ecuyer, Efficient and portable combined random number generators, Commun. ACM, 31, 6 (1988), 742-749 and 774.
Index entries for sequences related to pseudo-random numbers.

Cf. A384397, A384399, A384400, A384401, A384402.

Cf. A096550-A096561 other pseudo-random number generators.

NestList[Mod[42024*#, 2^31 - 847] &, 1, 50] (* Paolo Xausa, May 29 2025 *)

a(n) = 42024 * a(n-1) mod (2^31-847).

A384399 Consecutive states of the linear congruential pseudo-random number generator 41546*s mod (2^31-837) when started at s=1.

Original entry on oeis.org

1, 41546, 1726070116, 415531613, 62076069, 2032989474, 2081730174, 1986561601, 1628882794, 2086219292, 1660453472, 1609609959, 240622074, 352201199, 1750622311, 406689858, 2089566331, 1130153051, 774477142, 692384319, 266663829, 2099100496, 2099734917

Offset: 1

Sean A. Irvine, May 27 2025

Periodic with period 2^31-838.

Sean A. Irvine, Table of n, a(n) for n = 1..10000
Pierre L'Ecuyer, Efficient and portable combined random number generators, Commun. ACM, 31, 6 (1988), 742-749 and 774.
Index entries for sequences related to pseudo-random numbers.

Cf. A384397, A384398, A384400, A384401, A384402.

Cf. A096550-A096561 other pseudo-random number generators.

NestList[Mod[41546*#, 2^31 - 837] &, 1, 50] (* Paolo Xausa, May 30 2025 *)

a(n) = 41546 * a(n-1) mod (2^31-837).

cp's OEIS Frontend

A096550 Consecutive internal states of the IMSL pseudo-random number generator RNUN when started with ISEED=1.

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A384113 Consecutive states of a linear congruential pseudo-random number generator for MacModula-2 when started at 1.

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A384489 Consecutive states of the linear congruential pseudo-random number generator 392314069 * s mod 2^32 when started at s=1.

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A384158 Consecutive states of the linear congruential pseudo-random number generator for 16-bit WATFOR/WATFIV when started at 1.

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A384081 Consecutive internal states of a linear congruential pseudo-random number generator for the Hewlett-Packard HP-20S when started at 1.

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Mathematica

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A384126 Consecutive states of a linear congruential pseudo-random number generator (93*s+1) mod 2^13 when started at s=1.

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Maple

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A384289 Consecutive internal states of the linear congruential pseudo-random number generator for GWBASIC 3.23 when started at 1.

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