

A116084


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


3



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
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OFFSET

1,4


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 oneterm 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


LINKS

Table of n, a(n) for n=1..41.


FORMULA

A116085(n) = a(n+1)  a(n).
a(n) = Sum_{k=1..n1} A116085(k), cf. examples.  M. F. Hasler, Jul 14 2016


EXAMPLE

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.


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 *)


CROSSREFS

Cf. A115855, A000009, A038566, A038567, A116085.
Equals A154888(n)1.
Sequence in context: A306145 A143184 A309173 * A108925 A279026 A120549
Adjacent sequences: A116081 A116082 A116083 * A116085 A116086 A116087


KEYWORD

nonn


AUTHOR

Reinhard Zumkeller, Feb 04 2006


EXTENSIONS

a(24)a(34) from Don Reble, Jul 13 2016
a(35)a(41) from Giovanni Resta, Jul 15 2016


STATUS

approved



