A061364 -id:A061364 - 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.

A096561 Consecutive internal states of the second of the two linear congruential random number generators whose combined output is used in function RANDOM_NUMBER in version 8 of the Intel FORTRAN Compiler for Linux, using its intrinsic initialization.

Original entry on oeis.org

2147483398, 2147442707, 491644535, 44073136, 275411947, 1494571342, 367188984, 1612130085, 1622029567, 724872099, 810967243, 1649143122, 223185073, 139696145, 126975187, 29251410, 592572674, 1023646436, 1632766708, 1701483674, 1908878648, 1615402586, 1642669521

Offset: 1

Hugo Pfoertner, Aug 13 2004

This is part 2 of a combined pseudorandom number generator proposed by Pierre L'Ecuyer. For more information, references and links see A096560. For the spectral properties see Table 1, line 21, on page 106 of Knuth's TAOCP Vol. 2.

This sequence has period 2^31 - 250. - Charles R Greathouse IV, Sep 10 2015

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
Pierre L'Ecuyer, Efficient and portable combined random number generators, Commun. ACM, 31, 6 (1988), 742-749 and 774.
Index entries for periodic sequences with large period.
Index entries for sequences related to pseudo-random numbers.

Cf. A096550-A096560, A061364.

a:= proc(n) option remember; `if`(n=1, 2147483398,
      irem(40692 *a(n-1), 2147483399))
    end:
seq(a(n), n=1..30);  # Alois P. Heinz, Jun 10 2014

NestList[Mod[#*40692, 2^31 - 249] &, 2^31 - 250, 50] (* Paolo Xausa, Aug 29 2024 *)

a(n)=lift(-Mod(40692,2147483399)^(n-1)) \\ M. F. Hasler, May 14 2015

a(1)=2^31-250, a(n)=40692*a(n-1) mod (2^31-249).

A084277 Pseudo-random numbers: Davenport's generator for 32-bit integers.

Original entry on oeis.org

10904, 21520, 20895, 25213, 7963, 32668, 25961, 6826, 13530, 17404, 5948, 5449, 8262, 14606, 10788, 30720, 31758, 4871, 32173, 13874, 11466, 27821, 31595, 8091, 15213, 16731, 31046, 17581, 21096, 21170, 7302, 19629, 18847, 30239, 2465, 6541

Offset: 0

Frank Ellermann, May 25 2003

Davenport recommends b(n) as generator, a(n) uses bits 16..30 of b(n).

H. Davenport, The Higher Arithmetic, 7th ed. 1999, Cambridge University Press, ch. 8.3.

Index entries for sequences related to pseudo-random numbers.

Cf. A084276, A084277, A061364 (for other versions).

Cf. A382684 (b(n)).

#include 
unsigned next= 1; int i= 0;
int main() { while (i++ < 36) next = next * 2147001325 + 715136305, printf( "%d ", (next/65536) % 32768 ); }

b(n) = b(n-1) * 2147001325 + 715136305, b(0) = 1, a(n) = (( b(n)/(2^16) ) mod (2^15)).

A260083 Consecutive numbers generated by the 'random' function in PARI (32 bit, version > 2.4) using the default argument (2^31).

Original entry on oeis.org

326511477, 1362174608, 903267448, 656347688, 1853455872, 693790135, 1743812782, 1381849689, 1919384312, 561030593, 610902088, 1730365257, 1381380589, 1437333698, 1880143150, 1861285526, 135271255, 2062787134, 476812016, 748713098, 702275007, 1351923632

Offset: 1

Felix Fröhlich, Jul 15 2015

nonn

These are the values the function generates in PARI/GP v 2.7.0, using Richard Brent's XORGEN algorithm.
For all n, 0 < a(n) < 2^31.
All 32-bit PARI versions > 2.4 should yield this sequence of pseudo-random numbers, when the random generator is in its initial state or reset to that state using setrand(1). - M. F. Hasler, Sep 18 2016
The generator has a period of 2^4096-1. - Hugo Pfoertner, Sep 18 2016

Felix Fröhlich, Table of n, a(n) for n = 1..10000
Richard P. Brent, Some long-period random number generators using shifts and xors, ANZIAM Journal 48 (CTAC2006), C188-C202, 2007.
PARI, Catalogue of GP/PARI Functions, Conversions and similar elementary functions or commands: random({N = 2^{{31}}})
Index entries for sequences related to pseudo-random numbers.

Cf. A061364, A096558, A221555, A221556, A221557, A221558, A221559, A221560, A221561, A221562.

Cf. A276820: Corresponding 64 bit random number generator.

a(n) = random \\ Works only if called first with n=1, then with n=2, etc, and if the internal random generator was never called before or reset via setrand(1)

setrand(1); A260083=vector(100,i,random()) \\ M. F. Hasler, Sep 18 2016

A084276 Pseudo-random numbers: gcc 2.6.3 version for 32-bit integers.

Original entry on oeis.org

1, 7261, 13503, 15223, 8047, 88, 29253, 1572, 18740, 14060, 25029, 21260, 21203, 8054, 14922, 1084, 23230, 9538, 15465, 24014, 1274, 13434, 20285, 1973, 18297, 3025, 8994, 27214, 8752, 5869, 5683, 29094, 24317, 15766, 22147, 906, 13626, 21410, 32284, 13393, 27934

Offset: 0

Frank Ellermann, May 25 2003

Index entries for sequences related to pseudo-random numbers.

Cf. A084275, A084277, A061364 (other versions).

Cf. A385127 (internal state).

#include 
unsigned next= 1; int i= 0;
int main() { while (i++ < 37) next= next *69069 +5, printf( "%d ", (unsigned)( next >> 16) & 0x7FFF ); }

a(n) = floor(A385127(n + 1) / 2^16) mod 2^15. - Sean A. Irvine, Jun 18 2025

A096553 Consecutive states of the linear congruential pseudo-random number generator used in function rand() in the Standard C library (VAX C) when started at 1.

Original entry on oeis.org

1, 1103527590, 377401575, 662824084, 1147902781, 2035015474, 368800899, 1508029952, 486256185, 1062517886, 267834847, 180171308, 836760821, 595337866, 790425851, 2111915288, 1149758321, 1644289366, 1388290519, 1647418052, 1675546029

Offset: 1

Hugo Pfoertner, Jul 18 2004

nonn

This is also the sequence of internal states of the generator described in Kernighan and Ritchie, which produces output limited to 15bit, see A061364. - A-number corrected by Jean-Claude Arbaut, Oct 05 2015

Brian W Kernighan and Dennis M. Ritchie, The C Programming Language (Second Edition) Prentice Hall Software Series, 1988.

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harvey P. Dale)
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).

[n eq 1 select 1 else (1103515245 * Self(n-1) + 12345) mod (2^31): n in [1..25]]; // Vincenzo Librandi, Oct 06 2015

a:= proc(n) option remember; `if`(n<2, n,
      irem(1103515245 *a(n-1)+12345, 2147483648))
    end:
seq(a(n), n=1..30);  # Alois P. Heinz, Jun 10 2014

With[{c=2^31},NestList[Mod[1103515245#+12345,c]&,1,20]] (* Harvey P. Dale, Aug 01 2012 *)

a(n) = if(n<2, 1, (1103515245 * a(n-1) + 12345) % (2^31));
vector(100, n, a(n)) \\ Altug Alkan, Oct 05 2015

a(1)=1, a(n) = (1103515245 * a(n-1) + 12345) mod 2^31.

A357907 The internal state of the Sinclair ZX81 and Spectrum random number generator.

Original entry on oeis.org

1, 149, 11249, 57305, 38044, 35283, 24819, 26463, 18689, 25472, 9901, 21742, 57836, 12332, 7456, 34978, 1944, 14800, 61482, 23634, 3125, 37838, 19833, 45735, 22275, 32274, 61292, 9384, 48504, 33339, 10093, 36142, 23707, 8600, 55241, 14318, 25332, 64938, 20686, 44173, 36199, 27982

Offset: 1

Jacques Basaldúa, Oct 19 2022

The ZX81 had a congruential random number generator with the hardcoded values: x <- (75*x + 74) mod 65537.
This sequence starts with x = 1. The ZX81 had the option to start with a hardware counter.
The sequence has period 2^16. - Rémy Sigrist, Oct 20 2022
The ZX81 returned these values divided by 65536 as floating-point numbers, however, the seed was set as an integer using RAND (or RANDOMIZE on the ZX Spectrum). To produce the sequence as given here on the ZX81, set the seed with RAND 25340 (the last value in the period before it returns to 1), then print successive values with PRINT 65536*RND. On the ZX81, the current seed was stored in memory locations 16343 and 16384, and could be retrieved with PRINT 256*PEEK 16435+PEEK 16434 (which is equivalent to PRINT 65536*RND, but does not trigger stepping to the next value). - Sean A. Irvine, May 08 2025

Jianing Song, Table of n, a(n) for n = 1..65536
B. D. Ripley, Computer Generation of Random Variables: A Tutorial, International Statistical Review, 51 (1983), 301-309.
W. E. Sharp and Carter Bays, A review of portable random number generators, Computers and Geosciences, 18, 1 (1982), 79-87.
Wikipedia, Linear congruential generator.
Index entries for linear recurrences with constant coefficients, order 32769.
Index entries for sequences related to pseudo-random numbers.

Cf. A061364, A096550-A096561, A260083, A276820.

NestList[Mod[75*# + 74, 65537] &, 1, 50] (* Paolo Xausa, Oct 03 2024 *)

my(c=Mod(75,65537)); a(n) = lift(2*c^(n-1) - 1); \\ Kevin Ryde, Oct 22 2022

def a(n): return (2*pow(75, n-1, 65537) - 1)%65537
print([a(n) for n in range(1, 43)]) # Michael S. Branicky, Oct 23 2022

x <- 1
nxt <- function(x) (75*x + 74) %% 65537
for (t in 1:1000) {
  cat(sprintf('%i, ', x))
  x <- nxt(x)
}

a(n) = (75*a(n-1) + 74) mod 65537, a(1) = 1.
a(n + 2^16) = a(n). - Rémy Sigrist, Oct 20 2022
a(n) = (2*75^(n-1) - 1) mod 65537. - Kevin Ryde, Oct 20 2022
a(n) = a(n-1) - a(n-32768) + a(n-32769) for n > 32769. - Ray Chandler, Aug 03 2023

A084275 Pseudo-random numbers: MS C 6.0 version.

Original entry on oeis.org

41, 18467, 6334, 26500, 19169, 15724, 11478, 29358, 26962, 24464, 5705, 28145, 23281, 16827, 9961, 491, 2995, 11942, 4827, 5436, 32391, 14604, 3902, 153, 292, 12382, 17421, 18716, 19718, 19895, 5447, 21726, 14771

Offset: 0

Frank Ellermann, May 20 2003

Zhuorui He, Table of n, a(n) for n = 0..10000 (terms 0..98 from Alan Klette).
Sean A. Irvine, Java program (github)
Index entries for sequences related to pseudo-random numbers.

Cf. A084276, A084277, A096557, A096558, A061364, A384289.

#include 
int main()
{
    long next = 1;
    int i = 0;
    while ( i++ < 33 )
    {
        next = next * 214013L + 2531011L;
        printf( "%d ", (unsigned)( next >> 16) & 0x7FFF );
    }
    printf("\n");
    return 0;
}

a(n) = A096558(n+199670977) = floor((A096557(n+199670977) mod 2^31)/2^16). - Zhuorui He, Aug 22 2025

Corrected by Alan Klette, Jun 09 2019

A096554 Output of the linear congruential pseudo-random number generator used in function rand() as described in Kernighan and Ritchie, when seeded with 0.

Original entry on oeis.org

0, 21468, 9988, 22117, 3498, 16927, 16045, 19741, 12122, 8410, 12261, 27052, 5659, 9758, 21087, 25875, 32368, 26233, 15212, 17661, 20496, 8191, 23065, 23471, 32096, 10781, 14596, 23212, 24244, 5661, 514, 25643, 1350, 19576, 8051, 18234, 16882, 13023, 5983, 21166

Offset: 1

Hugo Pfoertner, Jul 18 2004

nonn

This sequence and A061364 are generated by the same algorithm but with different seed: A061364 (as well as A096553 which is the sequence of internal states of the generator) has seed 1, while this sequence has seed 0. [Corrected by Jean-Claude Arbaut, Oct 05 2015]

Brian W. Kernighan and Dennis M. Ritchie, The C Programming Language (Second Edition), Prentice Hall Software Series, 1988, page 46.

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

A061364 is the same generator seeded with 1.

static unsigned int next = 0; int rand( ) { next = next * 1103515245 + 12345; return ((next >>16) & 32767); }

BitShiftRight[NestList[Mod[#*1103515245 + 12345, 2^31] &, 12345, 50], 16] (* Paolo Xausa, Aug 29 2024 *)

x(n) = if(n<1, 0, (1103515245 * x(n-1) + 12345) % (2^31));
vector(100, n, floor(x(n)/2^16)) \\ Altug Alkan, Oct 05 2015

Correction of seed value and second formula from David Fifield, May 23 2024: (Start)
x(0) = 0, x(n) = (1103515245 * x(n-1) + 12345) mod 2^31, a(n) = floor(x(n)/2^16).
a(n) = A061364(n + 1212780038). (End)

Name amended (start at 0) by David Fifield, May 23 2024

A096559 Consecutive states of a linear congruential pseudo-random number generator that has the spectrally best primitive root for 2^31-1 as multiplier.

Original entry on oeis.org

1, 62089911, 847344462, 1061653656, 1954074819, 226824280, 953102500, 1452288378, 50913524, 2133871779, 1843965925, 427233754, 195855103, 1546822229, 1652729917, 1636805220, 217994169, 1312006067, 208869911, 310792805, 675992938, 1109700100, 855351136, 863373758

Offset: 1

Hugo Pfoertner, Aug 14 2004

nonn

The results of the spectral tests for this generator are given in line 18 of Table 1 in D. Knuth's TAOCP vol. 2, page 106.

G. A. Fishman, L. R. Moore III; An exhaustive analysis of multiplicative congruential random number generators with modulus 2^31-1. SIAM Journal on Scientific and Statistical Computing, Volume 7, Issue 1 (1986), 24-45. Erratum, ibid, Vol. 7, Issue 3 (1986) p. 1058.
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
Index entries for sequences related to pseudo-random numbers.

Cf. A096550-A096561, A061364.

a:= proc(n) option remember; `if`(n<2, n,
      irem(62089911 *a(n-1), 2147483647))
    end:
seq(a(n), n=1..30);  # Alois P. Heinz, Jun 10 2014

NestList[Mod[#*62089911, 2^31 - 1] &, 1, 50] (* Paolo Xausa, Aug 29 2024 *)

a(n)=lift(Mod(62089911,2147483647)^(n-1)) \\ M. F. Hasler, May 14 2015

a(1)=1, a(n)=62089911*a(n-1) mod (2^31-1).

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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A096561 Consecutive internal states of the second of the two linear congruential random number generators whose combined output is used in function RANDOM_NUMBER in version 8 of the Intel FORTRAN Compiler for Linux, using its intrinsic initialization.

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A084277 Pseudo-random numbers: Davenport's generator for 32-bit integers.

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C

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A260083 Consecutive numbers generated by the 'random' function in PARI (32 bit, version > 2.4) using the default argument (2^31).

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PARI

PARI

A084276 Pseudo-random numbers: gcc 2.6.3 version for 32-bit integers.

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C

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A096553 Consecutive states of the linear congruential pseudo-random number generator used in function rand() in the Standard C library (VAX C) when started at 1.

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Magma

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A357907 The internal state of the Sinclair ZX81 and Spectrum random number generator.

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