A154888 Number of ways to partition 1 into distinct reduced fractions i/j with j <= n.

1, 1, 2, 3, 5, 7, 11, 16, 24, 37, 48, 71, 88, 133, 284, 435, 472, 773, 826, 1835, 4369, 5546, 5649, 9924, 16465, 19944, 32324, 75913, 76168, 140802, 141141, 238514, 537697, 598296, 2556065, 4674085, 4674844, 4985386, 9716587, 23983712, 23984971, 48523606, 48525215

Offset: 1

Author

Reinhard Zumkeller, Jan 18 2009

Keywords

• nonn

Comments

a(n) = A116084(n) + 1 for all n because the decompositions are the same except for the additional fraction 1/1 allowed here but excluded in A116084. - M. F. Hasler, Jul 14 2016

Examples

			a(6) = #[1, 5/6+1/6, 4/5+1/5, 3/4+1/4, 2/3+1/3, 3/5+2/5, 1/2+1/3+1/6] = 7.
		

Crossrefs

Cf. A116085, A119983, A154887.

Equals A116084(n) + 1.

Programs

Mathematica

Farey[n_] := Union@ Flatten@ Table[ a/b, {b, n}, {a, b}]; t[n_, k_] := t[n, k] = Block[{c = j = 0, ip = IntegerPartitions[1, {k}, Farey@ n]}, len = 1 + Length@ ip; While[j < len, If[Plus @@ Union@ ip[[j]] == 1, c++ ]; j++ ]; c]; f[n_] := Plus @@ Table[ t[n, k], {k, Ceiling[n/2]}]; Array[f, 24] (* Robert G. Wilson v, Aug 30 2010 *)

Extensions

a(22)-a(26) from Robert G. Wilson v, Aug 30 2010

a(27)-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