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.

A116085 First differences of A116084.

Original entry on oeis.org

0, 1, 1, 2, 2, 4, 5, 8, 13, 11, 23, 17, 45, 151, 151, 37, 301, 53, 1009, 2534, 1177, 103, 4275, 6541, 3479, 12380, 43589, 255, 64634, 339, 97373, 299183, 60599, 1957769, 2118020, 759, 310542, 4731201, 14267125, 1259, 24538635, 1609, 57443858, 260450002, 8940128
Offset: 1

Views

Author

Reinhard Zumkeller, Feb 04 2006

Keywords

Comments

a(n-1) is the number of ways 1 can be written as sum of distinct positive fractions less than 1, having no denominator larger than n, and at least one equal to n (in its reduced form). (This follows from the definition of this sequence as first differences of A116084 or A154888, but these sequences are typically computed as partial sums of this one and could therefore be considered as less fundamental.) - M. F. Hasler, Jul 14 2016

Examples

			a(1) = 0 since there is no way to write 1 as sum of distinct fractions with denominator not larger than 2.
a(2) = # [1/3+2/3] = 1,
a(3) = # [1/4+3/4] = 1,
a(4) = # [1/5+4/5, 2/5+3/5] = 2,
a(5) = # [1/6+5/6, 1/6+1/3+1/2] = 2.

Crossrefs

Cf. A115856, A116084, A154888.

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 && MemberQ[Union@ Denominator@ #, n] &], {n, 2, 25}] (* Michael De Vlieger, Jul 15 2016 *)

Formula

a(n) = A116084(n+1) - A116084(n).
a(p-1) = A000009(p) - 1 for prime p.

Extensions

a(23)-a(40) from Giovanni Resta, Jul 15 2016
More terms from Jinyuan Wang, Dec 14 2024

A115855 Number of partitions of 1 into fractions i/j with 1<=i

Original entry on oeis.org

0, 1, 3, 6, 12, 21, 35, 58, 106, 188, 243, 493, 593, 1062, 3275, 5507, 5803, 12426, 12915, 42410, 131772, 167587, 168841, 428012, 839367, 1015501, 1968161, 5787286, 5791850, 15163758, 15170599, 28838712, 75983559, 82753547, 486356262, 1158442726, 1158464362
Offset: 1

Views

Author

Reinhard Zumkeller, Feb 01 2006

Keywords

Examples

			a(4) = #{1/2+1/2, 1/2+1/4+1/4, 1/3+2/3, 1/3+1/3+1/3, 1/4+3/4, 1/4+1/4+1/4+1/4} = 6.

Crossrefs

Cf. A000041, A002966, A038566, A038567, A115856, A116084.

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 &], {n, 25}] (* Michael De Vlieger, Jul 15 2016 *) (* or *)
a[n_] := Sum[ Length@ IntegerPartitions[1, {k}, Union@ Flatten[ Table[i/j, {j, n}, {i, j-1}]]], {k, n}]; Array[a, 20] (* Giovanni Resta, Jun 15 2017 *)

Formula

A115856(n) = a(n+1) - a(n).

Extensions

a(21)-a(28) from Michael De Vlieger, Jul 15 2016
More terms from Jinyuan Wang, Dec 11 2024

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
Showing 1-3 of 3 results.

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