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.

A119983 Number of ways to partition 1 into reduced fractions i/j with j <= n.

Original entry on oeis.org

1, 2, 4, 7, 13, 22, 36, 59, 107, 189, 244, 494, 594, 1063, 3276, 5508, 5804, 12427, 12916, 42411, 131773, 167588, 168842, 428013, 839368, 1015502, 1968162, 5787287, 5791851, 15163759, 15170600, 28838713, 75983560, 82753548, 486356263, 1158442727, 1158464363
Offset: 1

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Author

Franklin T. Adams-Watters, Aug 01 2006

Keywords

Comments

The reduced fractions are the Farey fractions of order n (A005728). - Robert G. Wilson v, Aug 30 2010

Examples

			a(3) = 4; 1 = 1/1 = 1/2 + 1/2 = 2/3 + 1/3 = 1/3 + 1/3 + 1/3.

Crossrefs

Cf. A000041, A020473, A115855 (one less), A115856.
Cf. A154886, A154888. - Reinhard Zumkeller, Jan 17 2009

Programs

  • Mathematica
    Farey[n_] := Union@ Flatten@ Table[a/b, {b, n}, {a, b}]; f[n_] := Length@ IntegerPartitions[1, All, Farey@ n]; Array[f, 27] (* Robert G. Wilson v, Aug 30 2010 *)

Formula

For p prime, a(p) = a(p-1) + P(p) - 1, where P is the partition function (A000041).

Extensions

Definition corrected by Reinhard Zumkeller, Jan 17 2009
a(21)-a(27) from Robert G. Wilson v, Aug 30 2010
More terms from Jinyuan Wang, Dec 12 2024

A154887 Number of ways to partition n into distinct reduced fractions i/j with j <= n.

Original entry on oeis.org

1, 2, 11, 71, 838, 7915, 181443, 3529287, 130501170, 4118232210, 269279551654, 9556917233108, 1003141976524301, 74252913818290142, 14979717449141067931, 1451159432555957095630, 363482056748832080145666, 28348494499719127795555178, 10422254792015005991605309232
Offset: 1

Views

Author

Reinhard Zumkeller, Jan 17 2009

Keywords

Examples

			a(3) = #{3, 8/3+1/3, 5/2+1/2, 7/3+2/3, 2+1, 2+2/3+1/3, 5/3+4/3, 5/3+1+1/3, 3/2+1+1/2, 3/2+2/3+1/2+1/3, 4/3+1+2/3} = 11. - corrected by _Reinhard Zumkeller_, Feb 02 2009

Crossrefs

Cf. A143270, A154886, A154888.

Extensions

More terms from Jinyuan Wang, Dec 13 2024

A290474 Number of fractional partitions of n where each element of a partition is a rational number r/s such that r < s, s <= n, and gcd(r,s) = 1.

Original entry on oeis.org

1, 0, 1, 12, 135, 4477, 100160, 8663934, 485380025, 80730951180, 10180011676356, 4126137351376215, 563950787766342780, 456369006693283278869, 200330760220853808357439, 335435016971402890883460861, 197675615401466868237710861644, 561969529551274362018496511765678
Offset: 0

Views

Author

Joseph Wheat, Aug 03 2017

Keywords

Comments

a(n) = (n^2 + 1)^(-1 + Sum_{k=1..n} phi(k)) - f(n) where phi(n) is Euler's totient function, and f(n) is the number of trivial solutions which do not satisfy the equation q_1*x_1 + q_2*x_2 + ... + q_m*x_m = n. Each coefficient is a rational number satisfying the criteria given in the definition, and m = -1 + Sum_{k=1..n} phi(k).

Examples

			For n=3, the number of partitions is equal to the number of nonnegative integer solutions for the equation: (1/2)*x_1 + (1/3)*x_2 + (2/3)*x_3 = 3. The set S of solutions is {[0,1,4], [0,3,3], [0,5,2], [0,7,1], [0,9,0], [2,0,3], [2,2,2], [2,4,1], [2,6,0], [4,1,1], [4,3,0], [6,0,0]}. Therefore, |S| = a(3) = 12.

Crossrefs

Cf. A015614, A057562, A154886, A154887, A171978.

Programs

  • PARI
    s(v, n, t) = {if(t==#v, f = n\v[t]; v[t]*f == n, sum(i=0, n\v[t], s(v, n-v[t]*i, t+1)))}
a(n) = {if(n<=2, return(n-1)); my(fractions = List(), q = 0); for(i=2, n, for(j=1, i-1, if(gcd(i, j)==1, listput(fractions, j/i)))); s(fractions, n, 1)} \\ David A. Corneth, Aug 03 2017

Extensions

a(7)-a(17) from Alois P. Heinz, Aug 03 2017
Showing 1-3 of 3 results.

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

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