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

Author

Reinhard Zumkeller, Feb 04 2006

Keywords

• nonn

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