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.

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A130266 A051340 * A128174.

Original entry on oeis.org

1, 1, 2, 4, 1, 3, 2, 5, 1, 4, 7, 2, 6, 1, 5, 3, 8, 2, 7, 1, 6, 10, 3, 9, 2, 8, 1, 7, 4, 11, 3, 10, 2, 9, 1, 8, 13, 4, 12, 3, 11, 2, 10, 1, 9, 5, 14, 4, 13, 3, 12, 2, 11, 1, 10, 16, 5, 15, 4, 14, 3, 13, 2, 12, 1, 11, 6, 17, 5, 16, 4, 15, 3, 14, 2, 13
Offset: 0

Views

Author

Gary W. Adamson, May 18 2007

Keywords

Comments

Row sums = A014255: (1, 3, 8, 12, 21, 27, 40, ...).
Left border = A123684: (1, 1, 4, 2, 7, 3, 10, 4, ...).

Examples

			First few rows of the triangle:
   1;
   1, 2;
   4, 1, 3;
   2, 5, 1, 4;
   7, 2, 6, 1, 5;
   3, 8, 2, 7, 1, 6;
  10, 3, 9, 2, 8, 1, 7;
  ...

Crossrefs

Cf. A051340, A128174, A014255, A123684.

Programs

  • Maple
    A128174 := proc(n,k)
    if k > n or k < 1 then
        0;
    else
        modp(k+n+1,2) ;
    end if;
end proc:
A051340 := proc(n,k)
    if k = n then
        n ;
    elif k <= n then
        1;
    else
        0;
    end if;
end proc:
A130266 := proc(n,k)
    add( A051340(n,j)*A128174(j,k),j=k..n) ;
end proc:
seq(seq(A130266(n,k),k=1..n),n=1..15) ; # R. J. Mathar, Aug 06 2016

Formula

A051340 * A128174 as infinite lower triangular matrices.

A337942 Table read by antidiagonals: T(n, k) is the least positive m such that (m*n) mod k = (m*k) mod n, with n > 0 and k > 0.

Original entry on oeis.org

1, 2, 2, 3, 1, 3, 4, 5, 5, 4, 5, 2, 1, 2, 5, 6, 3, 2, 2, 3, 6, 7, 3, 2, 1, 2, 3, 7, 8, 11, 2, 7, 7, 2, 11, 8, 9, 4, 17, 5, 1, 5, 17, 4, 9, 10, 5, 11, 13, 3, 3, 13, 11, 5, 10, 11, 5, 3, 2, 3, 1, 3, 2, 3, 5, 11, 12, 17, 7, 3, 28, 10, 10, 28, 3, 7, 17, 12
Offset: 1

Views

Author

Rémy Sigrist, Oct 01 2020

Keywords

Examples

			Array T(n, k) begins:
  n\k|  1   2   3   4   5   6   7   8   9  10
  ---+---------------------------------------
    1|  1   2   3   4   5   6   7   8   9  10
    2|  2   1   5   2   3   3  11   4   5   5
    3|  3   5   1   2   2   2  17  11   3   7
    4|  4   2   2   1   7   5  13   2   3   3
    5|  5   3   2   7   1   3   3  28  13   2
    6|  6   3   2   5   3   1  10   2   5   2
    7|  7  11  17  13   3  10   1   4   4  25
    8|  8   4  11   2  28   2   4   1  13   7
    9|  9   5   3   3  13   5   4  13   1   5
   10| 10   5   7   3   2   2  25   7   5   1

Links

Crossrefs

Cf. A123684.

Programs

  • PARI
    T(n,k) = for (m=1, oo, if ((m*n)%k==(m*k)%n, return (m)))

Formula

T(n, k) = T(k, n) <= k*n.
T(n, n) = 1.
T(n, 1) = n.
T(n, n^2) = n.
T(n, n^3) = n^2.
T(n, n^k) = n^(k-1) for any k > 0.
T(n, n+1) = A123684(n+1) for any n > 2.

A294689 Collatz cycle of negative numbers starting with -17.

Original entry on oeis.org

-17, -25, -37, -55, -82, -41, -61, -91, -136, -68, -34, -17, -25, -37, -55, -82, -41, -61, -91, -136, -68, -34, -17, -25, -37, -55, -82, -41, -61, -91, -136, -68, -34, -17, -25, -37, -55, -82, -41, -61, -91, -136, -68, -34
Offset: 1

Views

Author

Joe Slater, Nov 07 2017

Keywords

Comments

This is the longest of the three known cycles of negative numbers under the reduced Collatz transformation x: x=x/2 when x is even, and x=(3x+1)/2 when x is odd. The other two cycles of negative numbers are (-1,-1,-1,...) and (-5,-7,-10,-5,-7,-10,...). Only one cycle of positive numbers is known: (1,2,1,2,...). (0,0,0,...) is also a cycle.

Links

Crossrefs

Positive number transforms are in A014682; the absolute value of negative number transforms can be found in A123684.

Formula

a(1) = -17, a(n+1) = -A123684(abs(a(n))).
a(n+11) = a(n).
Previous Showing 11-13 of 13 results.

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