cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A127853 Numbers n such that A118680(n) = 1.

Original entry on oeis.org

10, 17, 26, 36, 37, 45, 50, 59, 61, 65, 67, 78, 82, 90, 91, 94, 101, 102, 105, 108, 110, 122, 136, 138, 145, 147, 149, 153, 155, 165, 170, 173, 181, 183, 188, 189, 193, 197, 210, 213, 220, 224, 226, 231, 232, 239, 249, 250, 257, 262, 263, 266, 268, 276, 279
Offset: 1

Views

Author

Alexander Adamchuk, Feb 03 2007

Keywords

Comments

Also a(n) are the numbers n such that 1 + Sum[ k, {k,1,n} ] = 1 + n(n+1)/2 divides Product[ k, {k,1,n} ] = n!. A118680[ a(n) ] = 1, where A118680(n) = {2, 2, 7, 11, 2, 11, 29, 37, 23, 1, 67, 79, 23, 53, 11, 137, 1, ...} = Absolute value of numerator of determinant of n X n matrix with M(i,j) = (i+1)/i if i=j otherwise 1. A118680(n) = Numerator[ (1 + n(n+1)/2) / n! ].

Crossrefs

Cf. A118680, A118679, A127852.

Programs

  • Mathematica
    Select[Range[1000],Numerator[(1 + #(#+1)/2)/#! ]==1&]

A118679 Absolute value of numerator of determinant of n X n matrix with M(i,j) = i/(i+1) if i=j otherwise 1.

Original entry on oeis.org

1, 2, 1, 13, 19, 13, 17, 43, 53, 1, 19, 89, 103, 59, 67, 151, 13, 47, 1, 229, 251, 137, 149, 1, 349, 47, 101, 433, 463, 1, 263, 43, 593, 157, 83, 701, 739, 389, 409, 859, 53, 59, 1, 1033, 83, 563, 587, 1223, 67, 331, 1, 1429, 1483, 769, 797, 127, 1709, 1, 457, 1889
Offset: 1

Views

Author

Alexander Adamchuk, May 19 2006, Feb 03 2007

Keywords

Comments

Numbers n such that a(n) = 1 are listed in A127852.
All a(n)>1 are prime belonging to A038889 (i.e., 17 is a square mod a(n)).

Crossrefs

Cf. A038889.
Cf. A118680, A127852, A127853.

Programs

  • Mathematica
    Numerator[Table[(-1)^(n+1) Det[ DiagonalMatrix[ Table[ i/(i+1) - 1, {i, 1, n} ] ] + 1 ], {n, 1, 70} ]]
Table[ Numerator[ (n^2+3n-2)/(2(n+1)!) ], {n,1,100} ]

Formula

det(M) = (-1)^(n+1)*(n^2+3*n-2)/(2*(n+1)!), implying that a(n)=p, where p=A006530(n^2+3*n-2) is the largest prime divisor of (n^2+3*n-2), if p>n+1 or p=sqrt((n^2+3*n-2)/2); otherwise a(n)=1.
a(n) = Numerator[(-1)^(n+1) Det[ DiagonalMatrix[ Table[ i/(i+1) - 1, {i, 1, n} ] ] + 1 ]].
a(n) = Numerator[ (n^2+3n-2)/(2(n+1)!) ] = Numerator[ ((2n+3)^2-17)/(4(n+1)!) ].

Extensions

Edited by Max Alekseyev, Jun 02 2009

A127852 Numbers n such that A118679(n) = 1.

Original entry on oeis.org

1, 3, 10, 19, 24, 30, 43, 51, 58, 62, 73, 75, 82, 94, 101, 106, 115, 116, 118, 128, 138, 147, 149, 159, 160, 163, 167, 172, 183, 186, 190, 191, 195, 201, 211, 214, 219, 249, 250, 252, 253, 260, 266, 272, 274, 277, 279, 283, 290, 294, 296, 306, 309, 310, 318
Offset: 1

Views

Author

Alexander Adamchuk, Feb 03 2007

Keywords

Comments

A118679[ a(n) ] = 1, where A118679(n) = {1, 2, 1, 13, 19, 13, 17, 43, 53, 1, 19, ...} = Absolute value of numerator of determinant of n X n matrix with M(i,j) = i/(i+1) if i=j otherwise 1. A118679(n) = Numerator[ (n^2+3n-2)/(2(n+1)!) ] = Numerator[ ((2n+3)^2-17)/(4(n+1)!) ].

Crossrefs

Cf. A118679, A118680, A127853.

Programs

  • Mathematica
    Select[Range[1000],Numerator[(#^2+3#-2)/(2(#+1)!)]==1&]

Formula

An integer n is in this sequence iff all prime divisors of n^2+3n-2 do not exceed n+1 and n^2+3n-2 is not of the form 2*p^2 for some prime p. [From Max Alekseyev, Jun 02 2009]
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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