A096561 -id:A096561 - cp's OEIS frontend

A096557 Consecutive internal states of the linear congruential pseudo-random number generator rand() used in Microsoft's Visual C++.

0, 2531011, 505908858, 3539360597, 159719620, 2727824503, 773150046, 548247209, 2115878600, 2832368235, 2006221698, 2531105853, 3989110284, 2222380191, 2165923046, 1345953809, 1043415696, 586225427, 3870123402, 2343900709, 3109564500, 3522190791, 2090033518, 3566711417

Offset: 1

Hugo Pfoertner, Jul 21 2004

nonn

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

Cf. A096558 corresponding output. A096550-A096561 other pseudo-random number generators.

a:= proc(n) option remember; `if`(n<2, 0,
      irem(214013 *a(n-1) +2531011, 4294967296))
    end:
seq(a(n), n=1..30);  # Alois P. Heinz, Jun 10 2014

NestList[Mod[#*214013 + 2531011, 2^32] &, 0, 50] (* Paolo Xausa, Aug 29 2024 *)

A096557(n)=lift((Mod(214013,2^34)^(n-1)-1)*13129821757)>>2 \\ M. F. Hasler, May 14 2015

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

A381318 Consecutive internal states of the linear congruential pseudo-random number generator (5^11*s+36643009) mod 2^30 when started at s=1.

Original entry on oeis.org

1, 85471134, 226025511, 290047084, 367499773, 919594538, 41492227, 69996632, 355863737, 216742518, 668535967, 32946180, 903885877, 903092354, 636577019, 361498480, 694791281, 514862158, 296723479, 683149980, 429969773, 598532570, 103941619, 668765576

Offset: 1

Sean A. Irvine, May 26 2025

Periodic with period 2^30.

The numbers on p. 124 of Grame and O'Donnel are given by s/2^30 using this generator starting from s=714133185.

Carl Grame and Dan O'Donnel, Learning BASIC: Programming Essentials, Science Research Associates, 1984 (see p. 124).

Sean A. Irvine, Table of n, a(n) for n = 1..10000
Index entries for sequences related to pseudo-random numbers.

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

Cf. A384388 (similar parameters).

NestList[Mod[5^11*# + 36643009, 2^30] &, 1, 50] (* Paolo Xausa, May 27 2025 *)

a(n) = (5^11 * a(n-1) + 36643009) mod 2^30.

A383940 Consecutive states of the linear congruential pseudo-random number generator (25173*s+13849) mod 2^16 when started at s=1.

Original entry on oeis.org

1, 39022, 61087, 20196, 45005, 3882, 21259, 65216, 19417, 30502, 20919, 26076, 16421, 44130, 63139, 32824, 14513, 51934, 36303, 35284, 8573, 11930, 41787, 65200, 9865, 29590, 743, 39628, 46037, 30162, 47315, 23080, 30049, 20814, 4351, 30916, 22317, 25098

Offset: 1

Sean A. Irvine, May 21 2025

Periodic with period 2^16.

This was a popular generator in the 1980's due to Grogono's book, but the period was too short for serious scientific use.

Peter Grogono, Programming in Pascal (2nd ed.), Addison-Wesley, 1984 (see pp. 136-137).

Sean A. Irvine, Table of n, a(n) for n = 1..10000
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 65536.

Cf. A096550-A096561 (other pseudo-random number generators).
Cf. A384082, A384085, A384150, A384194 (other early generators for Pascal).
Cf. A384220 (similar generator for Smalltalk-80).

NestList[Mod[25173*# + 13849, 2^16] &, 1, 100] (* Paolo Xausa, May 22 2025 *)

a(n) = (25173 * a(n-1) + 13849) mod 2^16.

A384114 Consecutive states of the linear congruential pseudo-random number generator (125*s+1) mod 2^12 when started at s=1.

Original entry on oeis.org

1, 126, 3463, 2796, 1341, 3786, 2211, 1944, 1337, 3286, 1151, 516, 3061, 1698, 3355, 1584, 1393, 2094, 3703, 28, 3501, 3450, 1171, 3016, 169, 646, 2927, 1332, 2661, 850, 3851, 2144, 1761, 3038, 2919, 332, 541, 2090, 3203, 3064, 2073, 1078, 3679, 1124, 1237

Offset: 1

Sean A. Irvine, May 19 2025

Periodic with period 4096.

W. F. Clocksin and C. S. Mellish, Programming in Prolog, Springer-Verlag, 1981 (see p. 149).

Sean A. Irvine, Table of n, a(n) for n = 1..4096
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 4096.

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

a:= proc(n) option remember; `if`(n<2, n,
      irem(125*a(n-1)+1, 2^12))
    end:
seq(a(n), n=1..45);  # Alois P. Heinz, May 19 2025

a[1] = 1; a[n_] := a[n] = Mod[125*a[n - 1] + 1, 2^12]; Array[a, 45, 1] (* Shenghui Yang, May 19 2025 *)

a(n) = (125*a(n-1) + 1) mod 2^12.

A384150 Consecutive states of the linear congruential pseudo-random number generator (10924*s+11830) mod (2^15+1) when started at s=1.

Original entry on oeis.org

1, 22754, 23661, 2722, 25475, 26382, 5443, 28196, 29103, 8164, 30917, 31824, 10885, 869, 1776, 13606, 3590, 4497, 16327, 6311, 7218, 19048, 9032, 9939, 21769, 11753, 12660, 24490, 14474, 15381, 27211, 17195, 18102, 29932, 19916, 20823, 32653, 22637, 23544

Offset: 1

Sean A. Irvine, May 21 2025

Periodic with period 32769.

This generator was given in Pascal by Richard Lamb in 1986. It was a reasonable generator for its time (with full period), but the period is way too small for modern use.

Richard Lamb, Pascal: Structure and Style, Benjamin-Cummings, 1986 (see pp. 226-227).

Sean A. Irvine, Table of n, a(n) for n = 1..10000
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 32769.

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

NestList[Mod[10924*# + 11830, 2^15 + 1] &, 1, 100] (* Paolo Xausa, May 22 2025 *)

my(f=Mod(10924, 10923*32769)); \
a(n) = lift(22753*f^((n-1) % 32769) - 11830) /10923; \\ Kevin Ryde, May 22 2025

a(n) = (10924 * a(n-1) + 11830) mod (2^15+1).

A384196 Consecutive states of the linear congruential pseudo-random number generator 20403*s mod 2^15 when started at s=1.

Original entry on oeis.org

1, 20403, 30505, 30891, 9361, 20579, 16953, 25819, 6689, 29715, 1609, 27659, 28849, 27331, 21337, 15931, 14401, 25715, 14697, 2923, 209, 4387, 18553, 923, 23137, 8403, 4233, 22219, 21745, 17283, 8601, 13563, 129, 10547, 2985, 20011, 27921, 483, 24249, 21083

Offset: 1

Sean A. Irvine, May 21 2025

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

Fred Maryanski, Digital Computer Simulation, Hayden Book Co., 1980 (see p. 224-230).

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(20403*a(n-1), 2^15))
    end:
seq(a(n), n=1..40);  # Alois P. Heinz, May 21 2025

NestList[Mod[20403*#, 2^15] &, 1, 100] (* Paolo Xausa, May 22 2025 *)

a(n) = lift(Mod(20403,32768)^(n-1)) \\ Jianing Song, Jul 06 2025

a(n) = 20403 * a(n-1) mod 2^15.

A384221 Consecutive states of the linear congruential pseudo-random number generator for the Texas Instruments TI99 when started at 1.

Original entry on oeis.org

1, 60062, 56335, 54564, 49133, 60602, 58139, 22240, 20761, 56598, 51559, 19676, 40837, 55218, 39667, 29464, 55089, 14478, 41919, 50580, 25629, 39850, 28619, 32848, 62025, 56582, 51991, 8012, 28085, 6306, 49571, 24200, 609, 43646, 40815, 14852, 7245, 11930

Offset: 1

Sean A. Irvine, May 22 2025

Periodic with period 2^16.

Sean A. Irvine, Table of n, a(n) for n = 1..10000
Lee Stewart, RND and RAND: Pseudorandom Number Generation by TI Basic, 2018.
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(28645*a(n-1)+31417, 2^16))
    end:
seq(a(n), n=1..38);  # Alois P. Heinz, May 22 2025

NestList[Mod[28645*# + 31417, 2^16] &, 1, 50] (* Paolo Xausa, May 23 2025 *)

a(n) = (28645 * a(n-1) + 31417) mod 2^16.

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.

A096555 Consecutive internal states of the linear congruential pseudo-random number generator RANDU that was used in the IBM Scientific Subroutine Library for IBM System/360 computers in the 1970's.

Original entry on oeis.org

1, 65539, 393225, 1769499, 7077969, 26542323, 95552217, 334432395, 1146624417, 1722371299, 14608041, 1766175739, 1875647473, 1800754131, 366148473, 1022489195, 692115265, 1392739779, 2127401289, 229749723, 1559239569, 845238963, 1775695897, 899541067, 153401569

Offset: 1

Hugo Pfoertner, Jul 19 2004

Due to a poor choice of the multiplier the generator fails most 3-d criteria for randomness. 9*a(n-2)-6*a(n-1)+a(n) = 0 mod 2^31. This was first described by George Marsaglia. The animated gif in the link demonstrates the deficient behavior. The animation shows 10000 points, each of whose coordinate triples (x,y,z) were formed from successive outputs of the generator. From a suitable view angle, it can be seen that these points do not fill the 3-D space, but lie in a few planes parallel to each other.

D. E. Knuth, The Art of Computer Programming Third Edition. Vol. 2 Seminumerical Algorithms. Chapter 3.3.4 The Spectral Test, Page 107. Addison-Wesley 1997.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Sarah Belet, 'Round the Twist, Blog Entry, Friday May 16 2014
George Marsaglia, Random numbers fall mainly in the planes, Proceedings of the National Academy of Sciences, 61, 25-28, 1968.
Hugo Pfoertner, Animation showing the deficient 3-d behavior, 2024.
Index entries for sequences related to pseudo-random numbers.
Index entries for linear recurrences with constant coefficients, order 536870912.

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

a:= proc(n) option remember; `if`(n<2, n,
      irem(65539 *a(n-1), 2147483648))
    end:
seq(a(n), n=1..30);  # Alois P. Heinz, Jun 10 2014

NestList[Mod[#*65539, 2^31] &, 1, 50] (* Paolo Xausa, Aug 29 2024 *)

a(n)=lift(Mod(65539,2^31)^(n-1)) \\ Charles R Greathouse IV, Jan 13 2016

a(1)=1, a(n) = 65539*a(n-1) mod 2^31. The sequence is periodic with period length 2^29.

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

cp's OEIS Frontend

A096557 Consecutive internal states of the linear congruential pseudo-random number generator rand() used in Microsoft's Visual C++.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A381318 Consecutive internal states of the linear congruential pseudo-random number generator (5^11*s+36643009) mod 2^30 when started at s=1.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Formula

A383940 Consecutive states of the linear congruential pseudo-random number generator (25173*s+13849) mod 2^16 when started at s=1.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Formula

A384114 Consecutive states of the linear congruential pseudo-random number generator (125*s+1) mod 2^12 when started at s=1.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A384150 Consecutive states of the linear congruential pseudo-random number generator (10924*s+11830) mod (2^15+1) when started at s=1.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A384196 Consecutive states of the linear congruential pseudo-random number generator 20403*s mod 2^15 when started at s=1.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A384221 Consecutive states of the linear congruential pseudo-random number generator for the Texas Instruments TI99 when started at 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple