A000043 -id:A000043 - cp's OEIS frontend

A249383 Smallest odd prime number Q such that Q*2^P-1 is also prime with P Mersenne prime exponent A000043(n).

3, 3, 7, 3, 31, 7, 61, 43, 79, 19, 739, 103, 2707, 4513, 139, 13, 4027, 28027, 14029, 87151, 11257, 8677, 122449, 104161, 113287, 216211, 150097, 862009, 876721, 414949, 4590451, 1391281

Offset: 1

Pierre CAMI, Oct 27 2014

nonn

a(32) is a 227838-digit certified prime.

Does a(n) exist for each n? - Charles R Greathouse IV, Oct 28 2014

Cf. A000043, A249382, A249384.

lista(nn) = {vmp = readvec("b000043.txt"); for (n=1, nn, k=2; while(!isprime(prime(k)*2^vmp[n]-1), k++); print1(prime(k), ", "););} \\ Michel Marcus, Oct 27 2014

A249384 Smallest odd prime number Q such that Q*2^P+1 is also prime, where P is a Mersenne prime exponent A000043(n).

Original entry on oeis.org

3, 5, 3, 5, 5, 53, 11, 239, 53, 191, 1229, 5, 233, 347, 1367, 9767, 2063, 89, 14009, 3329, 19991, 50849, 2711, 337871, 46301, 2543, 413093, 1157111, 615161, 1138649, 3778427

Offset: 1

Pierre CAMI, Oct 27 2014

Cf. A000043, A249382, A249383.

lista(nn) = {vmp = readvec("b000043.txt"); for (n=1, nn, k=2; while(!isprime(prime(k)*2^vmp[n]+1), k++); print1(prime(k), ", "););} \\ Michel Marcus, Oct 27 2014

A253027 Smallest odd number k>1 such that k*2^A000043(n)+1 is a prime number.

Original entry on oeis.org

3, 5, 3, 5, 5, 9, 11, 35, 53, 51, 105, 5, 233, 347, 125, 369, 2063, 89, 4715, 1145, 885, 4839, 2711, 30611, 5859, 2543, 21509, 114071, 309, 60191, 524489, 33305, 306363, 987537, 509765

Offset: 1

Pierre CAMI, Dec 26 2014

			3*2^2+1=13 prime so a(1)=3 as A000043(1)=2.
3*2^3+1=25 composite, 5*2^3+1=41 prime so a(2)=5 as A000043(2)=3.
3*2^5+1=97 prime so a(3)=3 as A000043(3)=5.

Cf. A000043, A135434.

a253027[n_] :=
Block[{k, t = Select[Prime[Range[n]], PrimeQ[2^# - 1] &], l},
  l = Length[t];
Table[k = 3; While[! PrimeQ[k*2^t[[i]] + 1], k = k + 2]; k, {i, l}]]; a253027[600] (* Michael De Vlieger, Dec 26 2014 *)

lista(nn) = {forprime (n=1, nn, if (isprime(2^n-1), k=3; while (!isprime(k*2^n+1), k += 2); print1(k, ", ");););} \\ Michel Marcus, Dec 27 2014

Command pfgw64 -f -e1000000 in.txt
in.txt file :
ABC2 a$*2^756839+$b // {number_primes,$b,1}
b: from 1 to 1
a: from 1 to 1000000

a(33)-a(35) from Pierre CAMI, Apr 06 2015

A059494 For odd p such that 2^p-1 is a prime (A000043), write 2^p-1 = x^2+3*y^2; sequence gives values of x.

Original entry on oeis.org

2, 2, 10, 46, 362, 298, 46162, 1505304098, 17376907720394, 9286834445316902, 9328321181472828398, 2107597973657165184339850860393713575649657317180489057212823189967494080057958, 22958222111004899714849436789827362390710508069726899926224050897274623732073762499062593658

Offset: 1

N. J. A. Sloane, Feb 05 2001

Representing a given prime P=3k+1 as x^2+3y^2 amounts to finding the shortest vector in a 2-dimensional lattice, namely either of the primes above P in the ring Q(sqrt(-3)). For instance, if P = 2^521 - 1 then P = x^2 + 3y^2 where x,y are 2107597973657165184339850860393713575649657317180489057212823189967494080057958, 898670952308059000662208200339860406351380028634597445743368513219427297854627. - Noam D. Elkies, Jun 25 2001

			p=7: 127 = 10^2 + 3*3^2, so a(3) = 10.

F. Lemmermeyer, Reciprocity Laws From Euler to Eisenstein, Springer-Verlag, 2000, p. 59.

Phil Moore, Tony Reix and others, Online Discussion

Cf. A000043, A000668, A059495.

f(p, P,a,m)= P=2^p-1; a=lift(sqrt(Mod(-3,P))); m=[P,a;0,1]; (m*qflll(m,1))~[1,]
for(n=1,11,print(abs(f([3,5,7,13,17,19,31,61,89,107,521][n])[1]))) \\ Joshua Zucker, May 23 2006

More terms from Noam D. Elkies, Jun 25 2001

Corrected and extended by Joshua Zucker, May 23 2006

A059495 For odd p such that 2^p-1 is a prime (A000043), write 2^p-1 = x^2+3*y^2; sequence gives values of y.

Original entry on oeis.org

1, 3, 3, 45, 3, 381, 2349, 115329357, 10279641655875, 5033685952807971, 5263826472436016979, 898670952308059000662208200339860406351380028634597445743368513219427297854627, 1163046333059983884005407440923089001865672520479660324278034104804471332063117356524700189

Offset: 1

N. J. A. Sloane, Feb 05 2001

			p=7: 127 = 10^2 + 3*3^2, so a(3) = 3.

F. Lemmermeyer, Reciprocity Laws From Euler to Eisenstein, Springer-Verlag, 2000, p. 59.

Phil Moore, Tony Reix and others, Online Discussion

Cf. A000043, A000668, A059494.

More terms from Joshua Zucker, May 23 2006

A099982 Bisection of A000043.

Original entry on oeis.org

2, 5, 13, 19, 61, 107, 521, 1279, 2281, 4253, 9689, 11213, 21701, 44497, 110503, 216091, 859433, 1398269, 3021377, 13466917, 24036583, 30402457, 37156667, 43112609

Offset: 1

N. J. A. Sloane, Nov 19 2004

Cf. A000043, A099983.

MersennePrimeExponent[Range[1, 48, 2]] (* Amiram Eldar, Oct 16 2024 *)

a(n) = A000043(2*n-1). - Amiram Eldar, Oct 16 2024

Offset corrected and a(20)-a(24) added by Amiram Eldar, Oct 16 2024

A099983 Bisection of A000043.

Original entry on oeis.org

3, 7, 17, 31, 89, 127, 607, 2203, 3217, 4423, 9941, 19937, 23209, 86243, 132049, 756839, 1257787, 2976221, 6972593, 20996011, 25964951, 32582657, 42643801, 57885161

Offset: 1

N. J. A. Sloane, Nov 19 2004

Cf. A000043, A099982.

MersennePrimeExponent[Range[2, 48, 2]] (* Amiram Eldar, Oct 16 2024 *)

a(n) = A000043(2*n). - Amiram Eldar, Oct 16 2024

Offset corrected and a(20)-a(24) added by Amiram Eldar, Oct 16 2024

A101416 Nearest k to j such that k*(2^j-1)-1 is prime where j=A000043(n) and 2^j-1 = Mersenne-prime(n) = A000668(n). If there are two k values equidistant from j, each of which produces a prime, the larger of the two gets added to the sequence.

Original entry on oeis.org

2, 2, 2, 6, 20, 14, 32, 90, 72, 80, 230, 80, 560, 740, 1542, 1782, 450, 828, 2562, 3936, 12474, 9288, 10224, 16022, 11088, 31034, 53972, 92372

Offset: 1

Pierre CAMI, Jan 16 2005

			n=7, j=A000043(7)=19, A000668(7)=524287, then k=6 or k=32 are the nearest values to j which produce primes so we take the larger of the two k values for a(7)=32.

Cf. A000043, A000668, A098555, A159585.

a(5)=20, a(20)=3936 corrected, other terms verified, a(27)-a(28) extended by Ray Chandler, Apr 16 2009

A107709 Least odd prime a(n) such that (a(n)M(n))^2 + a(n)M(n) - 1 is prime with M(n) = Mersenne-primes (A000043).

Original entry on oeis.org

3, 3, 3, 7, 43, 19, 13, 5, 571, 3, 137, 59, 3823, 2707, 6277, 1063, 4523, 631, 8209, 34537, 102329, 46399, 30323, 18803, 1063, 21019

Offset: 1

Pierre CAMI, Jun 10 2005

			M(1)=2^2-1=3, (3*3)^2 + 3*3 -1 = 89 prime so a(1)=3
M(2)=2^3-1=7, (3*7)^2 + 3*7 -1 = 461 prime so a(2)=3
M(3)=2^5-1=31, (3*31)^2 + 3*31 -1 = 8741 prime so a(3)=3
M(4)=2^7-1=127,(7*127)^2 + 7*127 -1 = 791209 prime so a(4)=7

Cf. A000043.

More terms from Pierre CAMI, Nov 21 2011

A113656 Digits of successive exponents of Mersenne primes (see A000043).

Original entry on oeis.org

2, 3, 5, 7, 1, 3, 1, 7, 1, 9, 3, 1, 6, 1, 8, 9, 1, 0, 7, 1, 2, 7, 5, 2, 1, 6, 0, 7, 1, 2, 7, 9, 2, 2, 0, 3, 2, 2, 8, 1, 3, 2, 1, 7, 4, 2, 5, 3, 4, 4, 2, 3, 9, 6, 8, 9, 9, 9, 4, 1, 1, 1, 2, 1, 3, 1, 9, 9, 3, 7, 2, 1, 7, 0, 1, 2, 3, 2, 0, 9, 4, 4, 4, 9, 7, 8, 6, 2, 4, 3, 1, 1, 0, 5, 0, 3, 1, 3, 2

Offset: 1

N. J. A. Sloane, following a suggestion of Bill McEachen, Jan 18 2006

Amiram Eldar, Table of n, a(n) for n = 1..235 (terms 1..187 from Bill McEachen, terms 188..227 from Gord Palameta)

Cf. A000043.

Flatten[IntegerDigits/@MersennePrimeExponent[Range[30]]] (* Harvey P. Dale, Aug 05 2021 *)

cp's OEIS Frontend

A249383 Smallest odd prime number Q such that Q*2^P-1 is also prime with P Mersenne prime exponent A000043(n).

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A249384 Smallest odd prime number Q such that Q*2^P+1 is also prime, where P is a Mersenne prime exponent A000043(n).

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Crossrefs

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PARI

A253027 Smallest odd number k>1 such that k*2^A000043(n)+1 is a prime number.

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PFGW

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A059494 For odd p such that 2^p-1 is a prime (A000043), write 2^p-1 = x^2+3*y^2; sequence gives values of x.

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PARI

Extensions

A059495 For odd p such that 2^p-1 is a prime (A000043), write 2^p-1 = x^2+3*y^2; sequence gives values of y.

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Author

Keywords

Examples

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Crossrefs

Extensions

A099982 Bisection of A000043.

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Author

Keywords

Crossrefs

Programs

Mathematica

Formula

Extensions

A099983 Bisection of A000043.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

Extensions

A101416 Nearest k to j such that k*(2^j-1)-1 is prime where j=A000043(n) and 2^j-1 = Mersenne-prime(n) = A000668(n). If there are two k values equidistant from j, each of which produces a prime, the larger of the two gets added to the sequence.

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Author

Keywords

Examples

Crossrefs

Extensions

A107709 Least odd prime a(n) such that (a(n)*M(n))^2 + a(n)*M(n) - 1 is prime with M(n) = Mersenne-primes (A000043).

Views

Author

Keywords

Examples

Crossrefs

Extensions

A107709 Least odd prime a(n) such that (a(n)M(n))^2 + a(n)M(n) - 1 is prime with M(n) = Mersenne-primes (A000043).