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

Views

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

A115855 Number of partitions of 1 into fractions i/j with 1<=i

Original entry on oeis.org

0, 1, 3, 6, 12, 21, 35, 58, 106, 188, 243, 493, 593, 1062, 3275, 5507, 5803, 12426, 12915, 42410, 131772, 167587, 168841, 428012, 839367, 1015501, 1968161, 5787286, 5791850, 15163758, 15170599, 28838712, 75983559, 82753547, 486356262, 1158442726, 1158464362
Offset: 1

Views

Author

Reinhard Zumkeller, Feb 01 2006

Keywords

Examples

			a(4) = #{1/2+1/2, 1/2+1/4+1/4, 1/3+2/3, 1/3+1/3+1/3, 1/4+3/4, 1/4+1/4+1/4+1/4} = 6.

Crossrefs

Cf. A000041, A002966, A038566, A038567, A115856, A116084.

Programs

  • Mathematica
    Table[Length@ Select[Flatten[Map[IntegerPartitions[1, {#}, Rest@ Union[Flatten@ TensorProduct[#, 1/#] &@ Range@ n /. {Integer -> 0, k /; k > 1 -> 0}]] &, Range@ n], 1], Total@ # == 1 &], {n, 25}] (* Michael De Vlieger, Jul 15 2016 *) (* or *)
a[n_] := Sum[ Length@ IntegerPartitions[1, {k}, Union@ Flatten[ Table[i/j, {j, n}, {i, j-1}]]], {k, n}]; Array[a, 20] (* Giovanni Resta, Jun 15 2017 *)

Formula

A115856(n) = a(n+1) - a(n).

Extensions

a(21)-a(28) from Michael De Vlieger, Jul 15 2016
More terms from Jinyuan Wang, Dec 11 2024

A116085 First differences of A116084.

Original entry on oeis.org

0, 1, 1, 2, 2, 4, 5, 8, 13, 11, 23, 17, 45, 151, 151, 37, 301, 53, 1009, 2534, 1177, 103, 4275, 6541, 3479, 12380, 43589, 255, 64634, 339, 97373, 299183, 60599, 1957769, 2118020, 759, 310542, 4731201, 14267125, 1259, 24538635, 1609, 57443858, 260450002, 8940128
Offset: 1

Views

Author

Reinhard Zumkeller, Feb 04 2006

Keywords

Comments

a(n-1) is the number of ways 1 can be written as sum of distinct positive fractions less than 1, having no denominator larger than n, and at least one equal to n (in its reduced form). (This follows from the definition of this sequence as first differences of A116084 or A154888, but these sequences are typically computed as partial sums of this one and could therefore be considered as less fundamental.) - M. F. Hasler, Jul 14 2016

Examples

			a(1) = 0 since there is no way to write 1 as sum of distinct fractions with denominator not larger than 2.
a(2) = # [1/3+2/3] = 1,
a(3) = # [1/4+3/4] = 1,
a(4) = # [1/5+4/5, 2/5+3/5] = 2,
a(5) = # [1/6+5/6, 1/6+1/3+1/2] = 2.

Crossrefs

Cf. A115856, A116084, A154888.

Programs

  • Mathematica
    Table[Length@ Select[Union /@ Flatten[Map[IntegerPartitions[1, {#}, Rest@ Union[Flatten@ TensorProduct[#, 1/#] &@ Range@ n /. {Integer -> 0, k /; k > 1 -> 0}]] &, Range@ n], 1], Total@ # == 1 && MemberQ[Union@ Denominator@ #, n] &], {n, 2, 25}] (* Michael De Vlieger, Jul 15 2016 *)

Formula

a(n) = A116084(n+1) - A116084(n).
a(p-1) = A000009(p) - 1 for prime p.

Extensions

a(23)-a(40) from Giovanni Resta, Jul 15 2016
More terms from Jinyuan Wang, Dec 14 2024
Showing 1-3 of 3 results.

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