A116085 First differences of A116084.

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

1,4

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

Links

Table of n, a(n) for n=1..45.

Formula

a(n) = A116084(n+1) - A116084(n).

a(p-1) = A000009(p) - 1 for prime p.

Example

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.

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 *)

Crossrefs

Cf. A115856, A116084, A154888.

Sequence in context: A079501 A093335 A093333 * A329692 A216198 A369708

Adjacent sequences: A116082 A116083 A116084 * A116086 A116087 A116088

Keyword

nonn,changed

Author

Reinhard Zumkeller, Feb 04 2006

Extensions

a(23)-a(40) from Giovanni Resta, Jul 15 2016

More terms from Jinyuan Wang, Dec 14 2024

Status

approved