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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A348020 a(n) is the minimum number of unit resistors in a circuit with resistance R = A007305(n)/A007306(n).

Original entry on oeis.org

1, 2, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 5, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 5, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8
Offset: 1

Views

Author

Hugo Pfoertner, Oct 14 2021

Keywords

Comments

For small values of n, the circuits are planar and correspond to the tiling of rectangles by squares. See A338573 for more information and examples.
The earliest nonplanar deviation occurs at a(3173) corresponding to R = 115/204 needing 11 instead of 12 resistors.

Links

Crossrefs

First 31 terms coincide with A070941.
Cf. A007305, A007306, A113881, A219158, A338573, A347282.
A338579 can be used for a lookup of the position for a given rational R.

A348051 Triangle T(j,k) of numerators of relativistically added fractional velocities w(u,v)=(u+v)/(u*v+1), with velocities enumerated by the Farey series, i.e., u(m) = v(m) = A007305(m)/A007306(m), m>=2.

Original entry on oeis.org

4, 5, 3, 7, 9, 12, 2, 7, 11, 8, 3, 11, 16, 13, 20, 11, 7, 19, 17, 25, 15, 10, 13, 17, 16, 23, 27, 24, 7, 1, 13, 3, 5, 5, 19, 5, 11, 13, 4, 1, 8, 31, 29, 17, 28, 14, 17, 5, 4, 31, 39, 36, 23, 37, 48, 13, 2, 23, 19, 29, 9, 33, 11, 7, 9, 21, 5, 19, 26, 23, 34, 41, 37, 9, 14, 53, 49, 56
Offset: 2

Views

Author

Hugo Pfoertner, Sep 25 2021

Keywords

Comments

The velocities are assumed to be given in units of c, and thus c = 1.

Examples

			The triangle of added fractions begins:
     u   1/2   1/3   2/3   1/4   2/5   3/5   3/4   1/5   2/7   3/8   3/7
   v \    .     .     .     .     .     .     .     .     .     .     .
  1/2 |  4/5    .     .     .     .     .     .     .     .     .     .
  1/3 |  5/7   3/5    .     .     .     .     .     .     .     .     .
  2/3 |  7/8   9/11 12/13   .     .     .     .     .     .     .     .
  1/4 |  2/3   7/13 11/14  8/17   .     .     .     .     .     .     .
  2/5 |  3/4  11/17 16/19 13/22 20/29   .     .     .     .     .     .
  3/5 | 11/13  7/9  19/21 17/23 25/31 15/17   .     .     .     .     .
  3/4 | 10/11 13/15 17/18 16/19 23/26 27/29 24/25   .     .     .     .
  1/5 |  7/11  1/2  13/17  3/7   5/9   5/7  19/23  5/13   .     .     .
  2/7 | 11/16 13/23  4/5   1/2   8/13 31/41 29/34 17/37 28/53   .     .
  3/8 | 14/19 17/27  5/6   4/7  31/46 39/49 36/41 23/43 37/62 48/73   .
  3/7 | 13/17  2/3  23/27 19/31 29/41  9/11 33/37 11/19  7/11  9/13 21/29

Links

Crossrefs

A348052 are the corresponding denominators.
Cf. A007305, A007306, A348131, A348132.

A348052 Triangle T(j,k) of denominators of relativistically added fractional velocities w(u,v)=(u+v)/(u*v+1), with velocities enumerated by the Farey series, i.e., u(m) = v(m) = A007305(m)/A007306(m), m>=2.

Original entry on oeis.org

5, 7, 5, 8, 11, 13, 3, 13, 14, 17, 4, 17, 19, 22, 29, 13, 9, 21, 23, 31, 17, 11, 15, 18, 19, 26, 29, 25, 11, 2, 17, 7, 9, 7, 23, 13, 16, 23, 5, 2, 13, 41, 34, 37, 53, 19, 27, 6, 7, 46, 49, 41, 43, 62, 73, 17, 3, 27, 31, 41, 11, 37, 19, 11, 13, 29, 6, 25, 29, 32, 43, 47, 40, 13, 19, 68, 61, 65
Offset: 2

Views

Author

Hugo Pfoertner, Sep 25 2021

Keywords

Examples

			See A348051.
The triangle starts:
  4/5,
  5/7,  3/5,
  7/8,  9/11, 12/13,
  2/3,  7/13, 11/14,  8/17,
  3/4, 11/17, 16/19, 13/22, 20/29
....

Crossrefs

A348051 are the corresponding numerators.
Cf. A007305, A007306, A348131, A348132.

A338579 Triangle T(D,N) read by rows, 1 <= N < D >= 2, where T(D,N) is the position of the fraction N/D in the Farey tree (or Stern-Brocot subtree) A007305/A007306.

Original entry on oeis.org

2, 3, 4, 5, 2, 8, 9, 6, 7, 16, 17, 3, 2, 4, 32, 33, 10, 12, 13, 15, 64, 65, 5, 11, 2, 14, 8, 128, 129, 18, 3, 24, 25, 4, 31, 256, 257, 9, 20, 6, 2, 7, 29, 16, 512, 513, 34, 19, 21, 48, 49, 28, 30, 63, 1024, 1025, 17, 5, 3, 23, 2, 26, 4, 8, 32, 2048
Offset: 2

Views

Author

Hugo Pfoertner, Nov 10 2020

Keywords

Comments

Fractions are reduced to lowest terms.

Examples

			The triangle begins
     N     1   2  3  4  5  6   7   8   9   10   11   12   13    14    15
   D \------------------------------------------------------------------
   2 |     2   .  .  .  .  .   .   .   .    .    .    .    .     .     .
   3 |     3   4  .  .  .  .   .   .   .    .    .    .    .     .     .
   4 |     5   2  8  .  .  .   .   .   .    .    .    .    .     .     .
   5 |     9   6  7 16  .  .   .   .   .    .    .    .    .     .     .
   6 |    17   3  2  4 32  .   .   .   .    .    .    .    .     .     .
   7 |    33  10 12 13 15 64   .   .   .    .    .    .    .     .     .
   8 |    65   5 11  2 14  8 128   .   .    .    .    .    .     .     .
   9 |   129  18  3 24 25  4  31 256   .    .    .    .    .     .     .
  10 |   257   9 20  6  2  7  29  16 512    .    .    .    .     .     .
  11 |   513  34 19 21 48 49  28  30  63 1024    .    .    .     .     .
  12 |  1025  17  5  3 23  2  26   4   8   32 2048    .    .     .     .
  13 |  2049  66 36 40 22 96  97  27  57   61  127 4096    .     .     .
  14 |  4097  33 35 10 41 12   2  13  56   15   62   64 8192     .     .
  15 |  8193 130  9 37  3  6 192 193   7    4   60   16  255 16384     .
  16 | 16385  65 68  5 80 11  47   2  50   14  113    8  125   128 32768
.
T(7,2) = 10 because A007306(10) = 7 and A007305(10) = 2 is the required double match, i.e., the position of the fraction 2/7 in the Farey tree is 10.
T(14,4) = T(7,2) = 10, because the fraction 4/14 reduced to lowest terms is 2/7.
T(16,12) = 8, because the fraction 12/16 reduced to lowest terms is 3/4, with the double match A007306(8)=4 and A007305(8)=3.

Links

Crossrefs

Cf. A007305, A007306, A054424, A054425.

Programs

  • PARI
    \\ using Yosu Yurramendi's formulas
a338579(lim)={
my(a7305=vectorsmall(2+2^(lim+2)),a7306=vectorsmall(2+2^(lim+2)));
  a7305[1]=1;
  for(m=1,lim,
     for(k=0,2^(m-1)-1,
      a7305[2^m+k]=a7305[2^(m-1)+k];
      a7305[2^m+2^(m-1)+k]=a7305[2^(m-1)+k]+a7305[2^m-k-1]
     )
  );
  a7306[1]=1;a7306[2]=2;
  for(m=0,lim,
     for(k=1,2^m,
      a7306[2^(m+1)+k]=a7306[2^m+k] + a7306[k];
      a7306[2^(m+1)-k+1]=a7306[2^m+k]
     )
  );
   my(findinFS(x)=for(k=2,#a7306,
      if(!(a7305[k-1]/a7306[k]-x),return(k)));0);
  for(de=2,lim+2,for(nu=1,de-1,my(q=nu/de);print1(findinFS(q),", ")))
};
a338579(10);
  • PARI
    T(d,n) = my(ret=1); d-=n; while(n!=d, ret<<=1; if(n>d, n-=d;ret++, d-=n)); ret+1; \\ Kevin Ryde, Nov 11 2020

A347282 a(n) is the least number of unit resistors in an electrical network with total resistance A007305(n)/A007306(n) such that all currents through the resistors are distinct and > 0.

Original entry on oeis.org

18, 19, 20, 18, 26, 19, 17, 19
Offset: 1

Views

Author

Hugo Pfoertner, Oct 12 2021

Keywords

Comments

The trivial case of a single resistor is excluded, thus making a(1) = A342558(18) = 18.

Examples

			  n  A007305(n)/ a(n)   Edge list of network graph (example)
     A007306(n)
  1     1/1       18    [[1, 5], [1, 6], [1, 7], [2, 5], [2, 7], [2, 8],
                         [3, 6], [3, 7], [3, 8], [3, 9], [10, 6], [4, 8],
                         [4, 9], [5, 8], [5, 9], [6, 9], [7, 9], [10, 4]]
                        Poles: [5, 10]
  2     1/2       19    see linked file for description of solutions
  3     1/3       20
  4     2/3       18
  5     1/4       26
  6     2/5       19
  7     3/5       17
  8     3/4       19
  9     1/5    unknown

Links

Crossrefs

Cf. A007305, A007306, A342558, A348020.

A049465 Replace each fraction p/q in Farey tree A007305/A007306 with 2p + q.

Original entry on oeis.org

1, 3, 4, 5, 7, 6, 9, 11, 10, 7, 11, 14, 13, 15, 18, 17, 13, 8, 13, 17, 16, 19, 23, 22, 17, 19, 26, 29, 25, 24, 27, 23, 16, 9, 15, 20, 19, 23, 28, 27, 21, 24, 33, 37, 32, 31, 35, 30, 21, 23, 34, 41, 37, 40, 47, 43, 32, 31, 41, 44, 37, 33, 36, 29, 19, 10, 17, 23, 22
Offset: 0

Views

Author

Gregory V. Richardson

Keywords

References

  • J. C. Lagarias, Number Theory and Dynamical Systems, pp. 35-72 of S. A. Burr, ed., The Unreasonable Effectiveness of Number Theory, Proc. Sympos. Appl. Math., 46 (1992). Amer. Math. Soc.

Formula

Equals 2*A007305 + A007306.

A049468 Replace each fraction p/q in Farey tree A007305/A007306 with p+2q.

Original entry on oeis.org

2, 3, 5, 7, 8, 9, 12, 13, 11, 11, 16, 19, 17, 18, 21, 19, 14, 13, 20, 25, 23, 26, 31, 29, 22, 23, 31, 34, 29, 27, 30, 25, 17, 15, 24, 31, 29, 34, 41, 39, 30, 33, 45, 50, 43, 41, 46, 39, 27, 28, 41, 49, 44, 47, 55, 50, 37, 35, 46, 49, 41, 36, 39, 31, 20, 17, 28, 37
Offset: 0

Views

Author

Gregory V. Richardson

Keywords

References

  • J. C. Lagarias, Number Theory and Dynamical Systems, pp. 35-72 of S. A. Burr, ed., The Unreasonable Effectiveness of Number Theory, Proc. Sympos. Appl. Math., 46 (1992). Amer. Math. Soc.

Crossrefs

Cf. A007305, A007306.

Formula

Equals A007305 + 2*A007306.

A049466 Replace each fraction p/q in Farey tree A007305/A007306 with 3p+q.

Original entry on oeis.org

1, 4, 5, 6, 9, 7, 11, 14, 13, 8, 13, 17, 16, 19, 23, 22, 17, 9, 15, 20, 19, 23, 28, 27, 21, 24, 33, 37, 32, 31, 35, 30, 21, 10, 17, 23, 22, 27, 33, 32, 25, 29, 40, 45, 39, 38, 43, 37, 26, 29, 43, 52, 47, 51, 60, 55, 41, 40, 53, 57, 48, 43, 47, 38, 25, 11, 19, 26, 25
Offset: 0

Views

Author

Gregory V. Richardson

Keywords

References

  • J. C. Lagarias, Number Theory and Dynamical Systems, pp. 35-72 of S. A. Burr, ed., The Unreasonable Effectiveness of Number Theory, Proc. Sympos. Appl. Math., 46 (1992). Amer. Math. Soc.

Formula

Equals 3*A007305+A007306.

A049467 Replace each fraction p/q in Farey tree A007305/A007306 with 4p+q.

Original entry on oeis.org

1, 5, 6, 7, 11, 8, 13, 17, 16, 9, 15, 20, 19, 23, 28, 27, 21, 10, 17, 23, 22, 27, 33, 32, 25, 29, 40, 45, 39, 38, 43, 37, 26, 11, 19, 26, 25, 31, 38, 37, 29, 34, 47, 53, 46, 45, 51, 44, 31, 35, 52, 63, 57, 62, 73, 67, 50, 49, 65, 70, 59, 53, 58, 47, 31, 12, 21, 29
Offset: 0

Views

Author

Gregory V. Richardson

Keywords

References

  • J. C. Lagarias, Number Theory and Dynamical Systems, pp. 35-72 of S. A. Burr, ed., The Unreasonable Effectiveness of Number Theory, Proc. Sympos. Appl. Math., 46 (1992). Amer. Math. Soc.

Formula

Equals 4*A007305+A007306.

A091765 Numbers n such that A007306(n) divides n.

Original entry on oeis.org

0, 1, 2, 3, 8, 18, 20, 56, 60, 80, 128, 135, 148, 205, 235, 255, 416, 434, 666, 1155, 1273, 1309, 1376, 1568, 1802, 2006, 2088, 2185, 2492, 2754, 2796, 2868, 3078, 3128, 3266, 3536, 3584, 3588, 3596, 3795, 3800, 3914, 3927, 4011, 4023, 4087, 4179, 4512, 4671
Offset: 1

Views

Author

Benoit Cloitre, Mar 06 2004

Keywords

Extensions

More terms from Michel Marcus, Dec 06 2013
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