A154888 -id:A154888 - cp's OEIS frontend

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

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

Franklin T. Adams-Watters, Aug 01 2006

nonn

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

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

Cf. A000041, A020473, A115855 (one less), A115856.

Cf. A154886, A154888. - Reinhard Zumkeller, Jan 17 2009

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 *)

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

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

A116084 Number of partitions of 1 into distinct fractions i/j with 1<=i

Original entry on oeis.org

0, 0, 1, 2, 4, 6, 10, 15, 23, 36, 47, 70, 87, 132, 283, 434, 471, 772, 825, 1834, 4368, 5545, 5648, 9923, 16464, 19943, 32323, 75912, 76167, 140801, 141140, 238513, 537696, 598295, 2556064, 4674084, 4674843, 4985385, 9716586, 23983711, 23984970, 48523605, 48525214

Offset: 1

Reinhard Zumkeller, Feb 04 2006

nonn

Partial sums of A116085, which is more elementary to compute, cf. examples. Sequence A154888 has an equivalent definition except that i=j is allowed there, which yields the one-term sum 1/1 as an additional possibility, and thus A154888(n) = a(n)+1. Sequence A115855 is also about the same problem but does not require the fractions to be distinct. - M. F. Hasler, Jul 14 2016

			a(4) = # [1/3+2/3, 1/4+3/4] = 2;
a(5) = a(4) + # [1/5+4/5, 2/5+3/5] = 2 + 2 = 4;
a(6) = a(5) + # [1/6+5/6, 1/6+1/3+1/2] = 4 + 2 = 6.

Cf. A000009, A038566, A038567, A115855, A116085.

Equals A154888(n) - 1.

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 &], {n, 25}] (* Michael De Vlieger, Jul 14 2016, after Robert G. Wilson v at A154888 *)

A116085(n) = a(n+1) - a(n).

a(n) = Sum_{k=1..n-1} A116085(k), cf. examples. - M. F. Hasler, Jul 14 2016

a(24)-a(34) from Don Reble, Jul 13 2016
a(35)-a(41) from Giovanni Resta, Jul 15 2016
a(42)-a(43) from Jinyuan Wang, Dec 12 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

Reinhard Zumkeller, Feb 04 2006

nonn

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

			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.

Cf. A115856, A116084, A154888.

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 *)

a(n) = A116084(n+1) - A116084(n).

a(p-1) = A000009(p) - 1 for prime p.

a(23)-a(40) from Giovanni Resta, Jul 15 2016

More terms from Jinyuan Wang, Dec 14 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

Reinhard Zumkeller, Jan 17 2009

nonn

			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

Cf. A143270, A154886, A154888.

More terms from Jinyuan Wang, Dec 13 2024

cp's OEIS Frontend

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

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A116084 Number of partitions of 1 into distinct fractions i/j with 1<=i

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A116085 First differences of A116084.

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

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