cp's OEIS Frontend

A116084 Number of partitions of 1 into distinct fractions i/j with 1<=i and i,j coprime.

%I A116084 #22 Dec 12 2024 11:08:56
%S A116084 0,0,1,2,4,6,10,15,23,36,47,70,87,132,283,434,471,772,825,1834,4368,
%T A116084 5545,5648,9923,16464,19943,32323,75912,76167,140801,141140,238513,
%U A116084 537696,598295,2556064,4674084,4674843,4985385,9716586,23983711,23984970,48523605,48525214
%N A116084 Number of partitions of 1 into distinct fractions i/j with 1<=i<j<=n and i,j coprime.
%C A116084 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
%F A116084 A116085(n) = a(n+1) - a(n).
%F A116084 a(n) = Sum_{k=1..n-1} A116085(k), cf. examples. - _M. F. Hasler_, Jul 14 2016
%e A116084 a(4) = # [1/3+2/3, 1/4+3/4] = 2;
%e A116084 a(5) = a(4) + # [1/5+4/5, 2/5+3/5] = 2 + 2 = 4;
%e A116084 a(6) = a(5) + # [1/6+5/6, 1/6+1/3+1/2] = 4 + 2 = 6.
%t A116084 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 *)
%Y A116084 Cf. A000009, A038566, A038567, A115855, A116085.
%Y A116084 Equals A154888(n) - 1.
%K A116084 nonn
%O A116084 1,4
%A A116084 _Reinhard Zumkeller_, Feb 04 2006
%E A116084 a(24)-a(34) from _Don Reble_, Jul 13 2016
%E A116084 a(35)-a(41) from _Giovanni Resta_, Jul 15 2016
%E A116084 a(42)-a(43) from _Jinyuan Wang_, Dec 12 2024