A096550 -id:A096550 - cp's OEIS frontend

A384400 Consecutive states of the linear congruential pseudo-random number generator 40692*s mod (2^31-249) when started at s=1.

1, 40692, 1655838864, 2103410263, 1872071452, 652912057, 1780294415, 535353314, 525453832, 1422611300, 1336516156, 498340277, 1924298326, 2007787254, 2020508212, 2118231989, 1554910725, 1123836963, 514716691, 445999725, 238604751, 532080813, 504813878

Offset: 1

Sean A. Irvine, May 27 2025

Periodic with period 2^31-250.

Implemented as gsl rng lecuyer21 in the GNU Scientific Library.

Sean A. Irvine, Table of n, a(n) for n = 1..10000
Mark Galassi, Jim Davies, James Theiler, Brian Gough, Gerard Jungman, Michael Booth, and Fabrice Rossi, GNU Scientific Library: Reference Manual, 2003 (see p. 184).
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, A384399, A384401, A384402.

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

NestList[Mod[40692*#, 2^31 - 249] &, 1, 50] (* Paolo Xausa, May 30 2025 *)

a(n) = 40692 * a(n-1) mod (2^31-249).

A384401 Consecutive states of the linear congruential pseudo-random number generator 40014*s mod (2^31-85) when started at s=1.

Original entry on oeis.org

1, 40014, 1601120196, 1346387765, 439883729, 732249858, 2127568003, 1962667596, 707287434, 1860990862, 1695805043, 1904850491, 53445315, 1814689225, 112933431, 612891482, 2124954851, 479214492, 407948861, 643161691, 28884682, 445508654, 322224693, 7553450

Offset: 1

Sean A. Irvine, May 27 2025

Periodic with period 2^31-86.

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, A384399, A384400, A384402.

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

NestList[Mod[40014*#, 2^31 - 85] &, 1, 50] (* Paolo Xausa, May 30 2025 *)

a(n) = 40014 * a(n-1) mod (2^31-85).

A384402 Consecutive states of the linear congruential pseudo-random number generator 39373*s mod (2^31-1) when started at s=1.

Original entry on oeis.org

1, 39373, 1550233129, 1548773083, 2044440394, 1622092461, 482805173, 2110316932, 1218777559, 1406738292, 1756031139, 1978020682, 2113853931, 894602131, 142925769, 1009147697, 429837187, 1808425391, 1165119911, 1868072236, 293238278, 798633422, 1138165032

Offset: 1

Sean A. Irvine, May 27 2025

Periodic with period 2^31-2.

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, A384399, A384400, A384401.

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

NestList[Mod[39373*#, 2^31 - 1] &, 1, 50] (* Paolo Xausa, May 30 2025 *)

a(n) = 39373 * a(n-1) mod (2^31-1).

A384746 Consecutive states of the linear congruential pseudo-random number generator MCNP from Los Alamos when started at 1.

Original entry on oeis.org

1, 19073486328125, 29763723208841, 187205367447973, 131230026111313, 264374031214925, 74735272014937, 31978779697717, 72377397341089, 127824407320157, 39323977335081, 168134765887429, 73951303845617, 27971537168493, 266449281326841, 41546074810965

Offset: 1

Sean A. Irvine, Jun 09 2025

Periodic with period 2^46.

Used for Monte Carlo simulations.

Sean A. Irvine, Table of n, a(n) for n = 1..10000
Forrest B. Brown and Yasunobu Nagaya, The MCNP5 Random Number Generator, LA-UR-02-3782, American Nuclear Society, Winter Meeting, 2002.
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.
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. A096550-A096561 other pseudo-random number generators.

a:= proc(n) option remember; `if`(n<2, n,
      irem(5^19*a(n-1), 2^48))
    end:
seq(a(n), n=1..16);  # Alois P. Heinz, Jun 09 2025

NestList[Mod[5^19*#, 2^48] &, 1, 30] (* Paolo Xausa, Jun 11 2025 *)

a(n) = 5^19 * a(n-1) mod 2^48.

A096558 Output of the linear congruential pseudo-random number generator rand() used in Microsoft's Visual C++.

Original entry on oeis.org

0, 38, 7719, 21238, 2437, 8855, 11797, 8365, 32285, 10450, 30612, 5853, 28100, 1142, 281, 20537, 15921, 8945, 26285, 2997, 14680, 20976, 31891, 21655, 25906, 18457, 1323, 28881, 2240, 9725, 32278, 2446, 590, 840, 18587, 16907, 21237, 23611, 12617

Offset: 1

Hugo Pfoertner, Jul 21 2004

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Index entries for sequences related to pseudo-random numbers.

Cf. A096557 corresponding internal states. A096550-A096561 other pseudo-random number generators.

b:= proc(n) option remember; `if`(n<2, 0,
      irem(214013 *b(n-1) +2531011, 4294967296))
    end:
a:= n-> irem(iquo(b(n), 65536), 32768):
seq(a(n), n=1..50);  # Alois P. Heinz, Jun 10 2014

A096557 = NestList[Mod[#*214013 + 2531011, 2^32] &, 0, 50];
Mod[BitShiftRight[A096557, 16], 2^15] (* Paolo Xausa, Aug 29 2024 *)

a(n)=A096557(n)>>16%2^15 \\ M. F. Hasler, May 14 2015

A096558(n)=lift((Mod(214013, 2^34)^(n-1)-1)*13129821757)>>18%32768 \\ M. F. Hasler, May 14 2015

b(1)=0, b(n)=(b(n-1) * 214013 + 2531011) mod 2^32. a(n) = (floor(b(n)/65536) mod 32768). The sequence is periodic with period length 2^32.

a(n) = floor(A096557(n)/2^16) mod 2^15 = floor((2531011*(214013^(n-1)-1)/214012 mod 2^32)/2^16) mod 2^15. - M. F. Hasler, May 14 2015

A384696 Consecutive states of the linear congruential pseudo-random number generator Cray RANF when started at 1.

Original entry on oeis.org

1, 44485709377909, 232253848878969, 94800993741645, 243522309605169, 20783065360997, 154093299791145, 161954398135485, 183663036741473, 207319719370837, 142356556532697, 278312552510253, 242082341486737, 37630394630981, 176334633251721, 233894773868189

Offset: 1

Sean A. Irvine, Jun 07 2025

Periodic with period 2^46.
Also implemented in the GNU Scientific Library as gsl_rng_ranf.
Also implemented in PASCLIB for Pascal programs on the CDC Cyber computers.

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.
Mark Galassi, Jim Davies, James Theiler, Brian Gough, Gerard Jungman, Michael Booth, and Fabrice Rossi, GNU Scientific Library: Reference Manual, 2003 (see p. 181).
B. D. Ripley, Computer Generation of Random Variables: A Tutorial, International Statistical Review, 51 (1983), 301-309.
Index entries for sequences related to pseudo-random numbers.

Cf. A384746, A384775, A384776, A384778, A384779, A384780.
Cf. A384546, A384547, A384548, A384549, A384550, A384551, A384552
Cf. A096550-A096561 other pseudo-random number generators.

a:= proc(n) option remember; `if`(n<2, n,
      irem(44485709377909*a(n-1), 2^48))
    end:
seq(a(n), n=1..16);  # Alois P. Heinz, Jun 09 2025

NestList[Mod[44485709377909*#, 2^48] &, 1, 30] (* Paolo Xausa, Jun 11 2025 *)

a(n) = 44485709377909 * a(n-1) mod 2^48.

A383809 Consecutive states of a linear congruential pseudo-random number generator for Lisp 1985 when started at 1.

Original entry on oeis.org

1, 17, 38, 144, 189, 201, 154, 108, 79, 88, 241, 81, 122, 66, 118, 249, 217, 175, 214, 124, 100, 194, 35, 93, 75, 20, 89, 7, 119, 15, 4, 68, 152, 74, 3, 51, 114, 181, 65, 101, 211, 73, 237, 13, 221, 243, 115, 198, 103, 245, 149, 23, 140, 121, 49, 80, 105, 28

Offset: 1

Sean A. Irvine, May 17 2025

An example of a terrible random number generator.

Periodic with period 125 (well below the modulus of 251).

Sean A. Irvine, Table of n, a(n) for n = 1..125
Richard P. Gabriel, Performance and Evaluation of Lisp Systems, MIT, 1985 (see p. 140).
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 125.

Cf. A001026.

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

a:= proc(n) option remember; `if`(n<2, n,
      irem(17*a(n-1), 251))
    end:
seq(a(n), n=1..58);  # Alois P. Heinz, May 21 2025

NestList[Mod[17*#, 251] &, 1, 100] (* Paolo Xausa, May 21 2025 *)

a(n) = 17 * a(n-1) mod 251.

A384152 Consecutive states of the linear congruential pseudo-random number generator used by OMNITAB II when started at 1.

Original entry on oeis.org

1, 125, 7433, 3429, 2641, 2445, 2521, 3829, 3489, 1949, 6057, 3461, 6641, 2733, 5753, 6421, 8001, 701, 5705, 421, 3473, 8141, 1817, 5941, 5345, 4573, 6377, 2501, 1329, 2285, 7097, 2389, 3713, 5373, 8073, 1509, 209, 1549, 5209, 3957, 3105, 3101, 2601, 5637, 113

Offset: 1

Sean A. Irvine, May 20 2025

Periodic with period length 2048.
A terrible generator with period much less than the modulus.
Also, RN5 of the IRCCRAND package.
Originally defined by Kruskal including the implementation s = 5*s mod 8192; s = 5*s mod 8192; s = 5*s mod 8192 (rather than s = 125*s mod 8192).

Sean A. Irvine, Table of n, a(n) for n = 1..2048
Edward J. Dudewicz, IRCCRAND - The Ohio State University Random Number Generator Package, Technical Report, 1974.
David Hogben, Sally T. Peavy, and Ruth N. Varner, OMNITAB II: User's Reference Manual, NBS Technical Note 552, 1971.
J. B. Kruskal, Extremely portable random number generator, Communications of the ACM, 12, 2 (1969), 93-94.
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 2048.

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

Cf. A383809, A384113, A384126, A384971, A384973 (other examples with short periods).

a:= proc(n) option remember; `if`(n<2, n,
      irem(125*a(n-1), 2^13))
    end:
seq(a(n), n=1..45);  # Alois P. Heinz, May 21 2025

NestList[Mod[125*#, 2^13] &, 1, 100] (* Paolo Xausa, May 21 2025 *)

a(n) = lift(Mod(5,8192)^(3*n-3)) \\ Jianing Song, Jul 06 2025

a(n) = 125 * a(n-1) mod 2^13.

A384160 Consecutive states of the linear congruential pseudo-random number generator for 36-bit WATFOR/WATFIV when started at 1.

Original entry on oeis.org

1, 131069, 17179082761, 17183408101, 34345582673, 53083917, 16988766937, 17848727413, 32066509217, 7739650845, 25740764841, 33596591109, 30610037745, 12186659885, 12166953849, 6296898965, 7334844225, 19577928253, 5497393481, 14152584229, 20226775953

Offset: 1

Sean A. Irvine, May 20 2025

Periodic with period 2^33 (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..10000
Index entries for sequences related to pseudo-random numbers.

Cf. A384158, A384159.

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

NestList[Mod[131069*#, 2^35] &, 1, 20] (* Stefano Spezia, May 24 2025 *)

a(n) = 131069 * a(n-1) mod 2^35.

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".

Original entry on oeis.org

1, 69069, 475559465, 2801775573, 1790562961, 3104832285, 4238970681, 2135332261, 381957665, 1744831853, 1303896393, 1945705589, 2707602097, 4198202557, 3820321881, 201201733, 2583294017, 4003049741, 2417848425, 1454463253, 3332335313, 2360275549, 2093206905, 2813570789

Offset: 1

Hugo Pfoertner, Jul 18 2004

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. 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.
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.

Alois P. Heinz, 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. A096550-A096561 (for other pseudo-random number generators).

Cf. A385127 (same multiplier).

a:= proc(n) option remember; `if`(n<2, n,
      irem(69069 *a(n-1), 4294967296))
    end:
seq(a(n), n=1..30);  # Alois P. Heinz, Jun 10 2014

NestList[Mod[#*69069, 2^32] &, 1, 50] (* Paolo Xausa, Aug 29 2024 *)

a(n)=lift(Mod(69069,2^32)^(n-1)) \\ Charles R Greathouse IV, Jan 14 2016

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

a(n) == 1 (mod 4). Hugo Pfoertner, Nov 21 2024

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A384400 Consecutive states of the linear congruential pseudo-random number generator 40692*s mod (2^31-249) when started at s=1.

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A384401 Consecutive states of the linear congruential pseudo-random number generator 40014*s mod (2^31-85) when started at s=1.

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A384402 Consecutive states of the linear congruential pseudo-random number generator 39373*s mod (2^31-1) when started at s=1.

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A384746 Consecutive states of the linear congruential pseudo-random number generator MCNP from Los Alamos when started at 1.

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A096558 Output of the linear congruential pseudo-random number generator rand() used in Microsoft's Visual C++.

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A384696 Consecutive states of the linear congruential pseudo-random number generator Cray RANF when started at 1.

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A383809 Consecutive states of a linear congruential pseudo-random number generator for Lisp 1985 when started at 1.

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A384152 Consecutive states of the linear congruential pseudo-random number generator used by OMNITAB II when started at 1.

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