A002968 -id:A002968 - cp's OEIS frontend

A005420 Largest prime factor of 2^n - 1.

3, 7, 5, 31, 7, 127, 17, 73, 31, 89, 13, 8191, 127, 151, 257, 131071, 73, 524287, 41, 337, 683, 178481, 241, 1801, 8191, 262657, 127, 2089, 331, 2147483647, 65537, 599479, 131071, 122921, 109, 616318177, 524287, 121369, 61681, 164511353, 5419

Offset: 2

N. J. A. Sloane

nonn

			2^6 - 1 = 63 = 3*21 = 9*7, so a(6) = 7.

J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Charles R Greathouse IV and Amiram Eldar, Table of n, a(n) for n = 2..1206 (terms up to 500 from T. D. Noe, terms 501..1000 from Charles R Greathouse IV, terms 1001..1206 from Amiram Eldar)
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
factordb.com, Status of 2^1207-1. The factorization of the composite factor C337 of 2^1207-1 with 337 decimal digits is considered by many to be the most desired open factorization problem.
R. K. Guy, Letter to G. B. Huff & N. J. A. Sloane, Aug 1974
S. S. Wagstaff, Jr., The Cunningham Project
Eric Weisstein's World of Mathematics, Mersenne Number

Cf. A000043, A000225, A006530, A002326, A002587.
Cf. similar sequences listed in A274906.
Cf. A337431 (a(n)=a(2n)), A359063 (a(n)=a(2n)=a(4n)), A359088.

[Maximum(PrimeDivisors(2^n-1)): n in [2..45]]; // Vincenzo Librandi, Jul 13 2016

a[n_] := a[n] = FactorInteger[2^n-1] // Last // First; Table[Print[{n, a[n]}, If[2^n-1 == a[n], " Mersenne prime", " "]]; a[n], {n, 2, 127}] (* Jean-François Alcover, Dec 11 2012 *)
Table[FactorInteger[2^n - 1][[-1, 1]], {n, 2, 40}] (* Vincenzo Librandi, Jul 13 2016 *)

for(n=2,44, v=factor(2^n-1)[,1]; print1(v[#v]", "));

a(n) = vecmax(factor(2^n-1)[,1]); \\ Michel Marcus, Dec 15 2022

a(n) = a(2n) iff a(n) > A002587(n). See A337431. - Thomas Ordowski, Jan 07 2014
a(n) = A006530(A000225(n)). - Vincenzo Librandi, Jul 13 2016
a(n) = 2^n-1 = A000225(n) iff n is a Mersenne exponent (A000043). - Bernard Schott, Dec 11 2022

Description corrected by Michael Somos, Feb 24 2002
More terms from Rick L. Shepherd, Aug 22 2002
Incorrect comments removed by Michel Marcus, Dec 15 2022

A272363 Number of ways to group the first 2n natural numbers into n pairs (xi,yi) with yi>xi, and such that the 2n numbers xi+yi and xi-yi are all different.

Original entry on oeis.org

1, 1, 0, 2, 12, 64, 220, 1886, 16346, 142420, 1302106, 14467384, 177079358

Offset: 0

Michel Marcus, Apr 27 2016

			For n=3, ((1,5), (2,3), (4,6)) is an instance of such grouping. ((2,3), (1,5), (4,6)) is considered to be the same grouping. The other one is ((1,3), (2,6), (4,5)). So a(3) = 2.

Mok-Kong Shen and Tsen-Pao Shen, Number theory Research Problem 39, Bull. Amer. Math. Soc. 68 (1962), 557.

Cf. A002968, A007631, A060963.

okperm(vp, n) = {for (k=1, n-1, if (vp[k] > vp[k+1], return (0));); for (k=1, n, if (vp[k+n] <= vp[k], return (0));); 1;}
a(n) = {nb = 0; nn = 2*n; for (j=0, nn!-1, vp = numtoperm(nn, j); if (okperm(vp, n), vs = vector(n, k, vp[k]+vp[k+n]); vd = vector(n, k, vp[k]-vp[k+n]); if (#vs + #vd == #Set(concat(vs, vd)), nb++););); nb;}

from sympy.utilities.iterables import multiset_partitions
def A272363(n):
    return 1 if n == 0 else sum(1 for p in multiset_partitions(list(range(1,2*n+1)),n) if max(len(d) for d in p) == 2 and len(set([sum(d) for d in p]))+len(set([abs(d[0]-d[1]) for d in p])) == 2*n) # Chai Wah Wu, Oct 08 2018

a(n) >= A002968(n). - Altug Alkan, Oct 05 2018

a(n) <= A060963(n). - Chai Wah Wu, Oct 08 2018

a(0), a(7)-a(10) from Alois P. Heinz, Oct 05 2018

a(11)-a(12) from Giovanni Resta, Oct 11 2018

A320129 Number of ways to group the first 2*n natural numbers into n pairs (xi, yi) such that the n numbers xi + yi are all different.

Original entry on oeis.org

1, 1, 2, 10, 55, 412, 3736, 40518, 505486, 7145031, 112844566, 1970286922, 37676184205

Offset: 0

Altug Alkan, Oct 06 2018

			For n = 2, a(2) = 2 since {(1,3), (2,4)} and {(1,2), (3,4)} are corresponding sets. {(2,4), (1,3)} and {(3,1), (4,2)} are considered to be the same grouping with {(1,3), (2,4)}.

Altug Alkan, Line plot of t(n)-t(n-1) where t(n) = a(n+1)/a(n) for n <= 11

Cf. A002968, A060963, A272363.

okperm(vp, n) = {for (k=1, n-1, if (vp[k] > vp[k+1], return (0)); ); for (k=1, n, if (vp[k+n] <= vp[k], return (0)); ); 1; }
a(n) = if(n==0, 1, {nb = 0; nn = 2*n; for (j=0, nn!-1, vp = numtoperm(nn, j); if (okperm(vp, n), vs = vector(n, k, vp[k]+vp[k+n]); if (#vs == #Set(concat(vs)), nb++); ); ); nb; } ) \\ after Michel Marcus at A272363

from sympy.utilities.iterables import multiset_partitions
def A320129(n):
    return 1 if n == 0 else sum(1 for p in multiset_partitions(list(range(1,2*n+1)),n) if max(len(d) for d in p) == 2 and len(set(sum(d) for d in p)) == n) # Chai Wah Wu, Oct 08 2018

a(n) >= A272363(n).

a(11)-a(12) from Giovanni Resta, Oct 09 2018

A007631 Number of solutions to non-attacking reflecting queens problem.

Original entry on oeis.org

1, 1, 0, 0, 2, 4, 0, 2, 10, 32, 38, 140, 496, 1186, 3178, 16792, 82038, 289566, 1139874, 5914118, 33800010, 142337180, 721286448, 4384569864

Offset: 0

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

a(n) is the number of ways to pair the natural numbers from 1 to n with those between n+1 and 2*n into n pairs (xi,yi) such that the 2*n numbers yi+i and yi-i are all different. - Michel Marcus, Apr 27 2016

			For n = 4, ((1,7), (2,5), (3,8), (4,6)) is an instance of such grouping. ((2,5), (1,7), (3,8), (4,6)) is considered to be the same grouping.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Jordan Bell, Brett Stevens, A survey of known results and research areas for n-queens, Discrete Mathematics, Volume 309 (2009), pp 1-31.
M. Gardner, Fractal Music, Hypercards and More, Freeman, NY, 1991, p. 240.
G. B. Huff, On pairings of the first 2n natural numbers, Acta. Arith. 23 (1973) 117-126.
D. A. Klarner, The Problem of Reflecting Queens, The American Mathematical Monthly, Vol. 74, No. 8 (Oct., 1967), pp. 953-955.
M. Slater, Number theory Research Problem 1, Bull. Amer. Math. Soc. 69 (1963), 333.

Cf. A000170, A002968, A051223, A051224, A272363.

a(n) = {nb = 0; for (j=0, n!-1, vp = numtoperm(n, j); vb = vector(n, k, vp[k]+n); vs = vector(n, k, vb[k]+k); vd = vector(n, k, vb[k]-k); if (#vs + #vd == #Set(concat(vs, vd)), nb++); ); nb; } \\ Michel Marcus,  Apr 27 2016

a(18)-a(21) from Sean A. Irvine, Jan 13 2018
a(0)-a(3) prepended by Michel Marcus, Oct 03 2018
a(22) from Sean A. Irvine, Oct 04 2018
a(23) from Sean A. Irvine, Oct 07 2018

A320168 Number of ways to group the first 2*n positive integers into n pairs (xi, yi) with xi < yi, and such that the n numbers (yi mod xi) are all different.

Original entry on oeis.org

1, 1, 2, 2, 7, 12, 22, 26, 85, 226, 717, 1695, 5071, 14275, 47405, 176747, 638329, 2166516

Offset: 0

Altug Alkan, Oct 07 2018

How does a(n+1)/a(n) behave as n increases?

			a(3) = 2 because {(1,3), (2,5), (4,6)} and {(1,5), (2,3), (4,6)} are corresponding sets.
a(4) = 7 because {(1,6), (2,5), (3,8), (4,7)}, {(1,3), (2,7), (4,6), (5,8)}, {(1,7), (2,3), (4,6), (5,8)}, {(1,3), (2,5), (4,7), (6,8)}, {(1,5), (2,3), (4,7), (6,8)}, {(1,2), (3,7), (4,6), (5,8)}, {(1,2), (3,8), (4,7), (5,6)} are corresponding sets.

David A. Corneth, Illustration for a(4)

Cf. A002968, A060963, A272363, A320129.

a(13)-a(17) from Rémy Sigrist, Oct 07 2018

cp's OEIS Frontend

A005420 Largest prime factor of 2^n - 1.

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A272363 Number of ways to group the first 2*n natural numbers into n pairs (xi,yi) with yi>xi, and such that the 2*n numbers xi+yi and xi-yi are all different.

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A320129 Number of ways to group the first 2*n natural numbers into n pairs (xi, yi) such that the n numbers xi + yi are all different.

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A007631 Number of solutions to non-attacking reflecting queens problem.

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A320168 Number of ways to group the first 2*n positive integers into n pairs (xi, yi) with xi < yi, and such that the n numbers (yi mod xi) are all different.

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A272363 Number of ways to group the first 2n natural numbers into n pairs (xi,yi) with yi>xi, and such that the 2n numbers xi+yi and xi-yi are all different.