A096550 -id:A096550 - cp's OEIS frontend

A221556 Consecutive values produced by the C++ minstd_rand random number generator with the default seed (1).

48271, 182605794, 1291394886, 1914720637, 2078669041, 407355683, 1105902161, 854716505, 564586691, 1596680831, 192302371, 1203428207, 1250328747, 1738531149, 1271135913, 1098894339, 1882556969, 2136927794, 1559527823, 2075782095, 638022372, 914937185, 1931656580

Offset: 1

Eric M. Schmidt, Jan 19 2013

This is a linear congruential random number generator with multiplier 48271.

Periodic with period 2^31-2. - Sean A. Irvine, May 30 2025

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from Eric M. Schmidt)
David G. Carta, Two Fast Implementations of the "Minimal Standard" Random Number Generator, Commun. ACM, 33, 1 (1990), 87-88.
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), pp. 1192-1201.
Index entries for sequences related to pseudo-random numbers.

Cf. A096550, A384405.

a:= proc(n) option remember; `if`(n=0, 1,
      irem(48271 *a(n-1), 2147483647))
    end:
seq(a(n), n=1..30);  # Alois P. Heinz, Oct 25 2017

f[n_] := PowerMod[48271, n, 2^31 -1]; Array[f, 23] (* Robert G. Wilson v, Nov 10 2014 *)

a(n) = 48271^n mod (2^31-1).

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

A382535 Consecutive states of Lehmer's original linear congruential pseudo-random number generator 23*s mod (10^8+1) when started at s=1.

Original entry on oeis.org

1, 23, 529, 12167, 279841, 6436343, 48035888, 4825413, 10984498, 52643452, 10799384, 48385830, 12874079, 96103815, 10387723, 38917627, 95105413, 87424478, 10762974, 47548400, 93613190, 53103349, 21377015, 91671341, 8440822, 94138905, 65194794, 99480248

Offset: 1

Sean A. Irvine, May 25 2025

Periodic with period 5882352.

This is the first linear congruential pseudo-random number generator described in the literature. As such, it is the forerunner of one of the most widely used techniques for generating pseudo-random numbers.

Sean A. Irvine, Table of n, a(n) for n = 1..10000
D. H. Lehmer, Mathematical methods in large-scale computing units, Proceedings of a Second Symposium on Large-Scale Digital Calculating Machinery (1949), 141-146.
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. A009967.

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

NestList[Mod[23*#, 10^8 + 1] &, 1, 50] (* Paolo Xausa, May 26 2025 *)

a(n) = 23 * a(n-1) mod (10^8+1).

A384159 Consecutive states of the linear congruential pseudo-random number generator for 32-bit WATFOR/WATFIV when started at 1.

Original entry on oeis.org

1, 20613, 424895769, 938169853, 404929649, 1693398709, 828374025, 631292077, 1220159969, 1976439269, 430365689, 2020481117, 2026879057, 763630101, 1799615721, 1993805069, 1909315521, 1935501125, 533477081, 1446792893, 636483633, 859521397, 574460361, 126586221

Offset: 1

Sean A. Irvine, May 20 2025

Periodic with period 2^29 (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, A384160.

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

NestList[Mod[20613*#, 2^31] &, 1, 23] (* Stefano Spezia, May 24 2025 *)

a(n) = 20613 * a(n-1) mod 2^31.

A384217 Consecutive states of the linear congruential pseudo-random number generator (843314861*s+453816693) mod 2^31 when started at s=1.

Original entry on oeis.org

1, 1297131554, 17103983, 1426780792, 2111429773, 1142766270, 888797147, 1081516660, 1471148505, 488941338, 1429379591, 2081849904, 166513637, 1928300854, 1776832243, 142642604, 236172977, 1916812562, 182141599, 551190760, 1397538365, 1487855278, 1455317259

Offset: 1

Sean A. Irvine, May 29 2025

Periodic with period 2^31 (Dyck et al. mistakenly give the period as 2^29).

Proposed by Dyck et al. for FORTRAN 77 on VAX or IBM computers.

V. A. Dyck, J. D. Lawson, and J. A. Smith, FORTRAN 77: An Introduction to Structured Problem Solving, Reston Pub. Co., 1984 (see p. 467).

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. A384387.

a:= proc(n) option remember; `if`(n<2, n,
      irem(843314861*a(n-1)+453816693, 2^31))
    end:
seq(a(n), n=1..23);  # Alois P. Heinz, May 29 2025

NestList[Mod[843314861*# + 453816693, 2^31] &, 1, 50] (* Paolo Xausa, May 30 2025 *)

a(n) = (843314861 * a(n-1) + 453816693) mod 2^31.

A384236 Consecutive states of the linear congruential pseudo-random number generator (9806*s+1) mod (2^17-1) when started at s=1.

Original entry on oeis.org

1, 9807, 92400, 111649, 125103, 66530, 52814, 32564, 33629, 122410, 4243, 57352, 99123, 108674, 50015, 110480, 65066, 114640, 94945, 33358, 86404, 34681, 83713, 123077, 122366, 97063, 93248, 38593, 40982, 5807, 58629, 38569, 67780, 120711, 120937, 108886

Offset: 1

Sean A. Irvine, May 22 2025

Periodic with period length 2^17-2.

In the way this generator was defined by Collins, it would return values in the range [1..2^17-1] rather than the more typical [0..2^17-2] (that is, 0 would instead be 2^17-1).

William J. Collins, Intermediate Pascal Programming, McGraw-Hill, 1986 (see p. 157).

Paolo Xausa, 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).

NestList[Mod[9806*#, 2^17 - 1] + 1 &, 1, 50] (* Paolo Xausa, May 23 2025 *)

a(n) = (9806 * a(n-1) mod (2^17-1)) + 1.

A384361 Consecutive internal states of the linear congruential pseudo-random number generator of the HP 48 series calculators when started at 999500333083533.

Original entry on oeis.org

999500333083533, 529199358633911, 43582181444437, 294922982088079, 41089642444893, 284830972469031, 786870433805477, 40703079813759, 869103111377453, 156083179654551, 561556952003317, 315753873725039, 722319935785213, 518159379358471, 201897051493957, 715330849773919

Offset: 1

Paolo Xausa, May 27 2025

The initial 999500333083533 seed is the one used by the calculators after a memory clean; successive executions of the RAND command give the terms of this sequence (divided by 10^15 and truncated to 12 significant digits).
See links for more information.

Paolo Xausa, Table of n, a(n) for n = 1..10000
John Meyers, Random thoughts on HP48 random functions, comp.sys.hp48 Google group, 1997.
Steve VanDevender and Detlef Mueller, How RAND Works, hpcalc.org, 1992.
Index entries for sequences related to pseudo-random numbers.

Cf. A384416 (starting at 1).

Cf. other pseudo-random number generators: A096550-A096561, A381318, A382535, A383809, A384081, A384221.

NestList[Mod[2851130928467*#, 10^15] &, 999500333083533, 15]

a(1) = 999500333083533; for n > 1, a(n) = 2851130928467*a(n-1) mod 10^15.

A384406 Consecutive internal states of the IMSL pseudo-random number generator RNUN when started with ISEED=1 and RNOPT=3.

Original entry on oeis.org

1, 397204094, 2083249653, 858616159, 557054349, 1979126465, 2081507258, 1166038895, 1141799280, 106931857, 142950581, 1759473232, 1125003378, 1832650327, 144277780, 2055193084, 638219178, 585429359, 1481612600, 2097586569, 486421192, 1477976737, 886403653

Offset: 1

Sean A. Irvine, May 27 2025

Periodic with period 2^31-2.

Also used by SAS and GPSS/PC.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Visual Numerics, IMSL Fortran Numerical Library: User's Guide, Stat Library, Version 7.0 (see p. 1430).
Index entries for sequences related to pseudo-random numbers.

Cf. A096550.

a:= proc(n) option remember; `if`(n<2, n,
      irem(397204094*a(n-1), 2^31-1))
    end:
seq(a(n), n=1..23);  # Alois P. Heinz, May 29 2025

NestList[Mod[397204094*#, 2^31 - 1] &, 1, 50] (* Paolo Xausa, May 30 2025 *)

a(n) = 397204094 * a(n-1) mod (2^31-1).

A384416 Consecutive internal states of the linear congruential pseudo-random number generator of the HP 48 series calculators when started at 1.

Original entry on oeis.org

1, 2851130928467, 261097470970089, 335429755623563, 468090732667921, 287888716607107, 194022960814969, 298923961822523, 84062462462241, 191517259514547, 165777802909449, 436661297384683, 996040654470961, 669370619746787, 188023750085529, 201468430854043, 677208350742081

Offset: 1

Paolo Xausa, May 28 2025

To initialize the seed to 1, use the RDZ command with an argument between 10^-16 and 10^-13 (for example, "1E-13 RDZ"). Successive executions of the RAND command give the terms of this sequence (divided by 10^15 and truncated to 12 significant digits).
After a memory clean, the calculators use the seed 999500333083533 (cf. A384361).
See the Meyers link for more information.
Periodic with period 10^14/2.

Paolo Xausa, Table of n, a(n) for n = 1..10000
John Meyers, Random thoughts on HP48 random functions, comp.sys.hp48 Google group, 1997.
Steve VanDevender and Detlef Mueller, How RAND Works, hpcalc.org, 1992.
Index entries for sequences related to pseudo-random numbers.

Cf. A384361 (starting at the default seed).

Cf. other pseudo-random number generators: A096550-A096561, A381318, A382535, A383809, A384081, A384221.

NestList[Mod[2851130928467*#, 10^15] &, 1, 20]

a(1) = 1; for n > 1, a(n) = 2851130928467*a(n-1) mod 10^15.

A384429 Consecutive states of the linear congruential pseudo-random number generator for Prime Sheffield Pascal when started at 1.

Original entry on oeis.org

1, 16807, 282475249, 1622647863, 947787489, 1578110407, 1878557649, 613813847, 2005365185, 1564292583, 1570623665, 602936439, 1724879009, 1159739911, 1187094929, 1381381783, 437908353, 499227175, 292517489, 751367351, 1027218017, 832165447, 1791151953

Offset: 1

Sean A. Irvine, May 28 2025

Periodic with period 2^28 (considerably less than the modulus).

A weak version of A096550.

J. R. Gilbert, The University of Sheffield Pascal System for Prime Computers, University of Sheffield, 1987 (see p. 10).

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.
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. A096550.

NestList[Mod[16807*#, 2^31] &, 1, 50] (* Paolo Xausa, May 30 2025 *)

a(n) = 16807 * a(n-1) mod 2^31.

cp's OEIS Frontend

A221556 Consecutive values produced by the C++ minstd_rand random number generator with the default seed (1).

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

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A382535 Consecutive states of Lehmer's original linear congruential pseudo-random number generator 23*s mod (10^8+1) when started at s=1.

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A384159 Consecutive states of the linear congruential pseudo-random number generator for 32-bit WATFOR/WATFIV when started at 1.

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A384217 Consecutive states of the linear congruential pseudo-random number generator (843314861*s+453816693) mod 2^31 when started at s=1.

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A384236 Consecutive states of the linear congruential pseudo-random number generator (9806*s+1) mod (2^17-1) when started at s=1.

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A384361 Consecutive internal states of the linear congruential pseudo-random number generator of the HP 48 series calculators when started at 999500333083533.

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A384406 Consecutive internal states of the IMSL pseudo-random number generator RNUN when started with ISEED=1 and RNOPT=3.