cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Original entry on oeis.org

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

Views

Author

Reinhard Zumkeller, Jan 18 2009

Keywords

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

A115856 First differences of A115855.

Original entry on oeis.org

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, 4564, 9371908, 6841, 13668113, 47144847, 6769988, 403602715, 672086464, 21636, 53588139, 1454972415, 6300092899, 44582
Offset: 1

Views

Author

Reinhard Zumkeller, Feb 01 2006

Keywords

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

Crossrefs

Cf. A116085, A119983.

Programs

  • 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 *)

Formula

a(n) = A115855(n+1) - A115855(n).
a(A000040(n)-1) = A000041(A000040(n)) - 1.

Extensions

a(20)-a(27) from Michael De Vlieger, Jul 15 2016
More terms from Jinyuan Wang, Dec 12 2024

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

Original entry on oeis.org

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

Views

Author

Reinhard Zumkeller, Feb 04 2006

Keywords

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
Showing 1-3 of 3 results.

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