A001348 -id:A001348 - cp's OEIS frontend

A129225 Residues of the Lucas - Lehmer primality test for M(29) = 536870911.

4, 14, 194, 37634, 342576132, 250734296, 433300702, 16341479, 49808751, 57936161, 211467447, 71320725, 91230447, 153832672, 217471443, 239636427, 223645010, 90243197, 27374393, 490737401, 35441039, 303927542, 202574536

Offset: 0

Sergio Pimentel, Apr 04 2007

Since a(27) > 0, M(29) = 536870911 is composite. Mersenne numbers are only prime if a(p-2) = 0.

			a(27) = 365171774^2 - 2 mod 536870911 = 458738443.

Dennis Martin, Table of n, a(n) for n = 0..27
Eric Weisstein's World of Mathematics, Lucas Lehmer Test.
Wikipedia, Lucas Lehmer Primality Test.

Cf. A095847, A003010, A129219, A129220, A129221, A129222, A129223, A129224, A129226, A001348.

NestList[Mod[#^2-2, 2^29-1] &, 4, 27] (* Ben Whitmore, Dec 28 2024 *)

a(0) = 4, a(n) = a(n-1)^2 - 2 mod 2^p-1, last term: a(p-2).

A234972 Least prime p < prime(n) such that 2^p - 1 is a primitive root modulo prime(n), or 0 if such a prime p does not exist.

Original entry on oeis.org

0, 0, 2, 2, 3, 3, 2, 2, 3, 2, 2, 17, 3, 2, 5, 2, 5, 3, 3, 3, 5, 2, 11, 2, 3, 2, 13, 3, 7, 2, 2, 5, 2, 2, 2, 3, 11, 2, 11, 2, 3, 7, 7, 7, 2, 2, 2, 2, 5, 3, 2, 3, 3, 7, 2, 3, 2, 11, 5, 2, 2, 2, 5, 5, 5, 2, 2, 5, 3, 3, 2, 3, 7, 7, 2, 7, 2, 3, 2, 7, 5, 31, 3, 3, 5, 3, 2, 5, 2, 2, 5, 5, 2, 3, 3, 5, 2, 2, 7, 7

Offset: 1

Zhi-Wei Sun, Apr 20 2014

nonn

Conjecture: a(n) > 0 for all n > 2.

			a(3) = 2 since 2 is a prime smaller than prime(3) = 5 with 2^2 - 1 = 3 a primitive root modulo prime(3) = 5.

Zhi-Wei Sun, Table of n, a(n) for n = 1..2000

Cf. A000040, A001348, A001918.

gp[g_,p_]:=Mod[g,p]>0&&(Length[Union[Table[Mod[g^k, p],{k,1,p-1}]]]==p-1)
Do[Do[If[gp[2^(Prime[k])-1,Prime[n]],Print[n," ",Prime[k]];Goto[aa]],{k,1,n-1}];Print[n," ",0];Label[aa];Continue,{n,1,100}]

A097743 Numbers of the form 3*2^(p - 1) - 1 where p is prime.

Original entry on oeis.org

5, 11, 47, 191, 3071, 12287, 196607, 786431, 12582911, 805306367, 3221225471, 206158430207, 3298534883327, 13194139533311, 211106232532991, 13510798882111487, 864691128455135231, 3458764513820540927

Offset: 1

Parthasarathy Nambi, Sep 21 2004

nonn

			p=2, 2^(2-1) + (2^2 - 1) = 5;
p=3, 2^(3-1) + (2^3 - 1) = 11.

Iain Fox, Table of n, a(n) for n = 1..467

Cf. A000396, A098102, A055010.

Table[3*2^(Prime[n] - 1) - 1, {n, 18}] (* Robert G. Wilson v, Sep 24 2004 *)

a(n) = 3*2^(prime(n) - 1) - 1; \\ Michel Marcus, Nov 30 2017

a(n) = A001348(n) + A061286(n). - Iain Fox, Dec 08 2017

Edited and extended by Robert G. Wilson v, Sep 24 2004

A129220 Residues of the Lucas - Lehmer primality test for M(11) = 2047.

Original entry on oeis.org

4, 14, 194, 788, 701, 119, 1877, 240, 282, 1736

Offset: 0

Sergio Pimentel, Apr 05 2007

Since a(9) > 0, M(11) is composite. In fact, 2047 = 23 * 89

			a(9) = a(8)^2 - 2 mod 2047 = 282^2 - 2 mod 2047 = 1736.

Eric Weisstein's World of Mathematics, Lucas Lehmer Test.
Wikipedia, Lucas Lehmer Primality Test.

Cf. A095847, A003010, A129218, A129219, A129221, A129222, A129223, A129224, A129225, A129226, A001348.

a(0) = 4; a(n) = a(n-1)^2-2 mod 2^p-1. Last term: a(p-2).

Offset corrected by Nathaniel Johnston, May 31 2011

A129221 Residues of the Lucas - Lehmer primality test for M(13) = 8191.

Original entry on oeis.org

4, 14, 194, 4870, 3953, 5970, 1857, 36, 1294, 3470, 128, 0

Offset: 0

Sergio Pimentel, Apr 04 2007

Since a(11) = 0, M(13) = 8191 is prime.

			a(11)= 128^2 - 2 mod 8191 = 16382 mod 8191 = 0

Eric Weisstein's World of Mathematics, Lucas Lehmer Test.
Wikipedia, Lucas Lehmer Primality Test.

Cf. A095847, A003010, A129219, A129220, A129222, A129223, A129224, A129225, A129226, A001348.

a(0) = 4 a(n) = a(n-1)^2 mod 2^p-1 Last term: a(p-2)

A129222 Residues of the Lucas - Lehmer primality test for M(17) = 131071.

Original entry on oeis.org

4, 14, 194, 37634, 95799, 119121, 66179, 53645, 122218, 126220, 70490, 69559, 99585, 78221, 130559, 0

Offset: 0

Sergio Pimentel, Apr 04 2007

Since a(15) = 0, M(17) = 131071 is prime.

			a(15) = 130559^2 - 2 mod 131071 = 0.

Eric Weisstein's World of Mathematics, Lucas Lehmer Test.
Wikipedia, Lucas Lehmer Primality Test.

Cf. A095847, A003010, A129219, A129220, A129221, A129223, A129224, A129225, A129226, A001348.

a(0) = 4, a(n) = a(n-1)^2 mod 2^p-1. Last term: a(p-2).

A129223 Residues of the Lucas - Lehmer primality test for M(19) = 524287.

Original entry on oeis.org

4, 14, 194, 37634, 218767, 510066, 386344, 323156, 218526, 504140, 103469, 417706, 307417, 382989, 275842, 85226, 523263, 0

Offset: 0

Sergio Pimentel, Apr 04 2007

Since a(17) = 0, M(19) = 524287 is prime.

			a(17) = 523263^2 - 2 mod 524287 = 0.

Eric Weisstein's World of Mathematics, Lucas Lehmer Test.
Wikipedia, Lucas Lehmer Primality Test.

Cf. A095847, A003010, A129219, A129220, A129221, A129222, A129224, A129225, A129226, A001348.

a(0) = 4, a(n) = a(n-1)^2 mod 2^p-1. Last term: a(p-2).

A129224 Residues of the Lucas - Lehmer primality test for M(23) = 8388607.

Original entry on oeis.org

4, 14, 194, 37634, 7031978, 7033660, 1176429, 7643358, 3179743, 2694768, 763525, 4182158, 7004001, 1531454, 5888805, 1140622, 4321431, 7041324, 2756392, 1280050, 6563009, 6107895

Offset: 0

Sergio Pimentel, Apr 04 2007

Since a(21) > 0, M(23) = 8388607 is composite. Mersenne numbers are only prime if a(p-2) = 0.

			a(21) = 6563009^2 - 2 mod 8388607 = 6107895.

Eric Weisstein's World of Mathematics, Lucas Lehmer Test.
Wikipedia, Lucas Lehmer Primality Test.

Cf. A095847, A003010, A129219, A129220, A129221, A129222, A129223, A129225, A129226, A001348.

a(0) = 4, a(n) = a(n-1)^2 mod 2^p-1. Last term: a(p-2).

A129226 Residues of the Lucas - Lehmer primality test for M(31) = 2147483647.

Original entry on oeis.org

4, 14, 194, 37634, 1416317954, 669670838, 1937259419, 425413602, 842014276, 12692426, 2044502122, 1119438707, 1190075270, 1450757861, 877666528, 630853853, 940321271, 512995887, 692931217, 1883625615, 1992425718

Offset: 0

Sergio Pimentel, Apr 04 2007

Since a(29) = 0, M(31) = 2147483647 is prime. Mersenne numbers are only prime if a(p-2) = 0.

			a(29) = 65536^2 - 2 mod 2147483647 = 0.

Dennis Martin, Table of n, a(n) for n = 0..29
Eric Weisstein's World of Mathematics, Lucas Lehmer Test.
Wikipedia, Lucas Lehmer Primality Test.

Cf. A095847, A003010, A129219, A129220, A129221, A129222, A129223, A129224, A129225, A001348.

p = 31; Mp = 2**p - 1
from itertools import accumulate
def f(anm1, _): return (anm1**2 - 2) % Mp
print(list(accumulate([4]*30, f))) # Michael S. Branicky, Apr 14 2021

a(0) = 4, a(n) = a(n-1)^2 mod 2^p-1. Last term: a(p-2).

A077586 a(n) = 2^(2^prime(n) - 1) - 1.

Original entry on oeis.org

7, 127, 2147483647, 170141183460469231731687303715884105727

Offset: 1

Henry Bottomley, Nov 07 2002

nonn

First four terms are primes. Fifth (1.61585...*10^616), sixth (5.45374...*10^2465), seventh (2.007...*10^39456) and eighth (1.298...*10^157826) are not primes.
Note that a(n) divides 2^a(n)-2 for every n, so if a(n) is composite then a(n) is a Fermat pseudoprime to base 2; cf. A007013. - Thomas Ordowski, Apr 08 2016
A number MM(p) is prime iff M(p) = A000225(p) = 2^p-1 is a Mersenne prime exponent (A000043), which isn't possible unless p itself is also in A000043. Primes of this form are called double Mersenne primes MM(p). For all Mersenne exponents between 7 and 61, factors of MM(p) are known. The next candidate MM(61) is far too large to be merely stored on any existing hard drive (it would require 3*10^17 bytes), but a distributed search for factors of this and other MM(p) is ongoing, see the doublemersenne.org web site. - M. F. Hasler, Mar 05 2020

			a(3) = 2^(2^5 - 1) - 1 = 2^31 - 1 = 2147483647.

Michael De Vlieger, Table of n, a(n) for n = 1..5
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
Luigi Morelli, DoubleMersennes.org
Eric Weisstein's World of Mathematics, Double Mersenne Number
Wikipedia, Double Mersenne number

Cf. A077585 (double Mersenne numbers), A000225 (Mersenne numbers), A001348 (ditto with prime indices), A000040 (primes).

A077586 := n -> 2^(2^ithprime(n)-1)-1; A077586(n) $ n=1..5; # M. F. Hasler, Mar 05 2020

Array[2^(2^Prime[#] - 1) - 1 &, 4] (* Michael De Vlieger, Apr 14 2022 *)

apply( {A077586(n)=2^(2^prime(n)-1)-1}, [1..5]) \\ M. F. Hasler, Mar 05 2020

a(n) = A077585(A000040(n)) = A000225(A001348(n)).

cp's OEIS Frontend

A129225 Residues of the Lucas - Lehmer primality test for M(29) = 536870911.

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A234972 Least prime p < prime(n) such that 2^p - 1 is a primitive root modulo prime(n), or 0 if such a prime p does not exist.

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A097743 Numbers of the form 3*2^(p - 1) - 1 where p is prime.

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A129220 Residues of the Lucas - Lehmer primality test for M(11) = 2047.

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A129221 Residues of the Lucas - Lehmer primality test for M(13) = 8191.

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A129222 Residues of the Lucas - Lehmer primality test for M(17) = 131071.

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A129223 Residues of the Lucas - Lehmer primality test for M(19) = 524287.

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A129224 Residues of the Lucas - Lehmer primality test for M(23) = 8388607.

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