A116085 -id:A116085

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A116084

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

+10
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

(list; graph; refs; listen; history; text; internal format)

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 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

LINKS

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

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

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.

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

A115856

First differences of A115855.

+10
2

1, 2, 3, 6, 9, 14, 23, 48, 82, 55, 250, 100, 469, 2213, 2232, 296, 6623, 489, 29495, 89362, 35815, 1254, 259171, 411355, 176134, 952660, 3819125

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

By definition of A115855, a(n+1) is the number of ways 1 can be written as sum of distinct positive fractions having no denominator larger than n, and at least one equal to n (in its reduced form). - M. F. Hasler, Jul 14 2016

LINKS

Table of n, a(n) for n=1..27.

FORMULA

a(n) = A115855(n+1) - A115855(n);

a(A000040(n)-1) = A000041(A000040(n))-1.

MATHEMATICA

Table[Length@ Select[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 *)

CROSSREFS

Cf. A116085.

KEYWORD

nonn,more

AUTHOR

Reinhard Zumkeller, Feb 01 2006

EXTENSIONS

More terms from Michael De Vlieger, Jul 15 2016

STATUS

approved

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