A095847 -id:A095847 - cp's OEIS frontend

A131460 Residues of 3^(2^(p(n)-1)+1) for Mersenne numbers with prime indices.

0, 5, 22, 118, 1803, 8182, 131062, 524278, 498820, 271127480, 2147483638, 44060320367, 967030303245, 7907414671310, 49672464783624, 5545884378065500, 125222315103997360, 2305843009213693942, 130613131595363896897

Offset: 1

Dennis Martin (dennis.martin(AT)dptechnology.com), Jul 13 2007, Jul 20 2007

nonn

Mp is prime iff 3^(2^(p(n)-1)+1) is congruent to (-9) Mod Mp. Thus M7 = 127 is prime because 3^65 Mod 127 = 118 (=127-9) while M11 = 2047 is composite because 3^1025 Mod 2047 <> 2038.

			a(5) = 3^(2^(11-1)+1) Mod 2^11-1 = 3^1025 Mod 2047 = 1803

Dennis Martin, Table of n, a(n) for n = 1..100

Cf. A095847, A001348, A131458, A131459, A131461, A131462, A131463.

a(n) = 3^(2^(p(n)-1)+1) Mod 2^p(n)-1

A131461 Residues of 3^(2^p(n)-2) for Mersenne numbers with prime indices.

Original entry on oeis.org

0, 1, 1, 1, 1013, 1, 1, 1, 5884965, 65165529, 1, 103888408793, 474639880182, 4112907695371, 72685811469476, 5155089749987738, 440411515280180314, 1, 95591506202441271281, 69291880649932219827

Offset: 1

Dennis Martin (dennis.martin(AT)dptechnology.com), Jul 20 2007

nonn

M_p is prime iff 3^(M_p-1) is congruent to 1 mod M_p. Thus M_7 = 127 is prime because 3^126 mod 127 = 1 while M_11 = 2047 is composite because 3^2046 mod 2047 <> 1.

			a(5) = 3^(2^11-2) mod 2^11-1 = 3^2046 mod 2047 = 1013

Dennis Martin, Table of n, a(n) for n = 1..100

Cf. A095847, A001348, A131458, A131459, A131460, A131462, A131463.

a(n) = 3^(2^p(n)-2) mod 2^p(n)-1

A131462 Residues of 3^(2^p(n)-1) for Mersenne numbers with prime indices.

Original entry on oeis.org

0, 3, 3, 3, 992, 3, 3, 3, 877681, 195496587, 3, 36787319437, 1423919640546, 3542630063906, 77319946053101, 6458069995222223, 168313041233693968, 3, 139200566017647400916, 207875641949796659481

Offset: 1

Dennis Martin (dennis.martin(AT)dptechnology.com), Jul 20 2007

nonn

M_p is prime iff 3 ^ M_p is congruent to 3 mod M_p. Thus M_7 = 127 is prime because 3^127 mod 127 = 3 while M_11 = 2047 is composite because 3^2047 mod 2047 <> 3.

			a(5) = 3^(2^11-1) mod 2^11-1 = 3^2047 mod 2047 = 992

Dennis Martin, Table of n, a(n) for n = 1..100

Cf. A095847, A001348, A131458, A131459, A131460, A131461, A131463.

a(n) = 3^(2^p(n)-1) mod 2^p(n)-1

A131463 Residues of 3^(2^p(n)) for Mersenne numbers with prime indices.

Original entry on oeis.org

0, 2, 9, 9, 929, 9, 9, 9, 2633043, 49618850, 9, 110361958311, 2072735666087, 1831797169511, 91222349803976, 1359811476184687, 504939123701081904, 9, 122453792873589376894, 623626925849389978443

Offset: 1

Dennis Martin (dennis.martin(AT)dptechnology.com), Jul 20 2007

nonn

M_p is prime iff 3^(M_p+1) is congruent to 9 mod M_p. Thus M_7 = 127 is prime because 3^128 mod 127 = 9 while M_11 = 2047 is composite because 3^2048 mod 2047 <> 9.

			a(5) = 3^(2^11) mod 2^11-1 = 3^2048 mod 2047 = 929

Dennis Martin, Table of n, a(n) for n = 1..100

Cf. A095847, A001348, A131458, A131459, A131460, A131461, A131462.

a(n) = 3^(2^p(n)) mod 2^p(n)-1

A135927 a(n) = a(n-1)^2 - 2 with a(1) = 10.

Original entry on oeis.org

10, 98, 9602, 92198402, 8500545331353602, 72259270930397519221389558374402, 5221402235392591963136699520829303150191924374488750728808857602

Offset: 1

Ant King, Dec 07 2007

This is the Lucas-Lehmer sequence with starting value u(1) = 10 and the position of the zeros when it is reduced mod(2^p - 1) also gives the position of the Mersenne primes. As we have started with n = 1, these will occupy the (p - 1)th positions in the sequence. For example, the first 12 terms mod(2^13 - 1) are 10, 98, 1411, 506, 2113, 672, 1077, 4996, 2037, 4721, 128, 0 and hence 8191 is a Mersenne prime. The radicals in the above closed forms are the solutions to x^2 - 10x + 1 = 0.

			a(4) = 2*cosh(2^3*log(5 + 2*sqrt(6))) = 92198402.

Gabriel Klambauer, Summation of Series, Amer. Math. Monthly, Vol. 87, No. 2 (Feb., 1980), pp. 128-130.
Raphael M. Robinson, Mersenne and Fermat Numbers, Proceedings of the American Mathematical Society, Vol. 5, No. 5. (October 1954), pp. 842-846.
E. L. Roettger and H. C. Williams, Some Remarks Concerning the Lucas-Lehmer Primality Test, Journal of Integer Sequences, Vol. 28 (2025), Article 25.2.5. See p. 3.
Wikipedia, Engel expansion

Cf. A000668, A000043, A003010, A095847, A001566, A135928, A246723.

a[1] = 10; a[n_] := a[n] = a[n - 1]^2 - 2; a[#]&/@Range[7]

A135927 = [10]
for n in range(1, 8): A135927.append(A135927[-1]**2-2)
print(A135927) # Karl-Heinz Hofmann, Feb 01 2022

a(n) = 2*cosh(2^(n-1)*log(5 + 2*sqrt(6))) = exp(2^(n-1)*log(5 + 2*sqrt(6))) + exp(2^(n-1)*log(5 - 2*sqrt(6))) = (5 + 2*sqrt(6))^(2^(n-1)) + (5 - 2*sqrt(6))^(2^(n-1)) = ceiling(exp(2^(n-1)*log(5 + 2*sqrt(6)))) = ceiling((5 + 2*sqrt(6))^(2^(n-1))).
From Peter Bala, Feb 01 2022: (Start)
Product_{n >= 1} (1 + 2/a(n)) = (1/2)*sqrt(6); Product_{n >= 1} (1 - 1/a(n)) = (4/11)*sqrt(6).
Engel expansion of 5 - sqrt(24) = 1/a(1) + 1/(a(1)*a(2)) + 1/(a(1)*a(2)*a(3)) + .... See Klambauer, p. 130. (End)

A317977 a(n) = A003010(n-2) mod (2^n - 1).

Original entry on oeis.org

1, 0, 14, 0, 23, 0, 149, 205, 95, 1736, 779, 0, 4193, 20400, 25439, 0, 221468, 0, 1036394, 840107, 1751891, 6107895, 5639594, 8772568, 66322529, 60611448, 99083624, 458738443, 989927528, 0, 3038229779, 5238898821, 393215, 11960838285, 27264928469, 117093979072, 274827575393, 276971366821

Offset: 2

Thomas Ordowski, Aug 12 2018

nonn

For n > 2, the Mersenne number 2^n - 1 is a prime if and only if a(n) = 0. See comments in A003010.

Chai Wah Wu, Table of n, a(n) for n = 2..3322

Cf. A000043, A000225, A000668, A003010, A095847.

a(n) = {my(pow = 2^n-1, res = Mod(4, pow)); for(i = 1, n-2, res = res^2 - 2); lift(res)}
first(n) = vector(n, i, a(i+1)) \\ David A. Corneth, Aug 12 2018

def A317977(n):
    m = 2**n-1
    c = 4 % m
    for _ in range(n-2):
        c = (c**2-2) % m
    return c # Chai Wah Wu, Oct 08 2018

a(prime(n)) = A095847(n).

More terms from Michel Marcus and David A. Corneth, Aug 12 2018

A331038 Residues of the Lucas-Lehmer primality test for M(127) = 2^127 - 1.

Original entry on oeis.org

3, 7, 47, 2207, 4870847, 23725150497407, 562882766124611619513723647, 9932388036497706472820043948129789713, 102423269049837077051675109560558766898, 7949236499829405891753012242872011683, 119093374737774941856311333667076322210

Offset: 0

Sergio Pimentel, Jan 08 2020

Since a(125) = 0, 2^127 - 1 = 170141183460469231731687303715884105727 is prime. This calculation was carried out by hand by Edouard Lucas. It took him 19 years from 1857 to 1876. The method works with a(0) = 3 since M(127) == 3 (mod 4). It also works with a(0) = 4 or a(0) = 10.

Sergio Pimentel, Table of n,a(n) for n = 0..125
Eric Weisstein's World of Mathematics, Lucas Lehmer Test.
Wikipedia, Lucas Lehmer Primality Test.

Cf. A000043, A000668, A001566, A095847.

Cf. also A129219, A129220, A129221, A129222, A129223, A129224, A129225.

NestList[Mod[#^2-2,2^127-1]&, 3,10] (* Stefano Spezia, Mar 28 2025 *)

a(n) = (a(n-1)^2 - 2) mod (2^127-1) with a(0) = 3; a(125) is the final term.

A248755 a(n) is the number of iterations for the Lucas-Lehmer sequence A003010 (mod p_n) to enter a loop, where p_n is the n-th prime number A000040(n).

Original entry on oeis.org

2, 2, 1, 4, 3, 3, 4, 2, 5, 4, 6, 5, 4, 5, 11, 3, 15, 6, 5, 3, 5, 6, 11, 13, 5, 4, 9, 27, 11, 10, 8, 7, 23, 13, 20, 12, 14, 10, 41, 28, 12, 4, 36, 4, 15, 13, 27, 8, 15, 11, 13, 24, 5, 51, 8, 65, 36, 8, 13, 47, 36, 42, 31, 20, 13, 52, 42, 6, 87, 16, 30, 89, 15, 7, 36, 95, 6, 17, 34, 10

Offset: 1

Robert G. Wilson v, Oct 13 2014

The Lucas-Lehmer sequence is used to test for Mersenne primes (A001348), but this is irrelevant for this sequence.

			a(4) is 4 because p_4 = 7, and the sequence A003010 (mod 7) becomes -> 4, 0, 5, 2, 2, 2, 2, 2, 2, .... The term 2 which is the first term of an infinite loop is at position 4.

GIMPS, Great Internet Mersenne Prime Search
Eric W. Weisstein's World of Mathematics, Lucas-Lehmer Test
Wikipedia, Lucas-Lehmer primality test

Cf. A003010, A001348, A095847, A000043.

f[n_] := -1 + Length@ NestWhileList[ Mod[#^2 - 2, Prime[n]] &, 4, UnsameQ[##] &, {2, Infinity}]; Array[f, 80]

cp's OEIS Frontend

A131460 Residues of 3^(2^(p(n)-1)+1) for Mersenne numbers with prime indices.

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A131461 Residues of 3^(2^p(n)-2) for Mersenne numbers with prime indices.

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A131462 Residues of 3^(2^p(n)-1) for Mersenne numbers with prime indices.

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A131463 Residues of 3^(2^p(n)) for Mersenne numbers with prime indices.

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A135927 a(n) = a(n-1)^2 - 2 with a(1) = 10.

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A317977 a(n) = A003010(n-2) mod (2^n - 1).

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A331038 Residues of the Lucas-Lehmer primality test for M(127) = 2^127 - 1.

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A248755 a(n) is the number of iterations for the Lucas-Lehmer sequence A003010 (mod p_n) to enter a loop, where p_n is the n-th prime number A000040(n).

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