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

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

Author

Reinhard Zumkeller, Feb 04 2006

Keywords

• nonn

Comments

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

Examples

			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.
		

Crossrefs

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

Equals A154888(n) - 1.

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

Formula

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

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

Extensions

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