

A116085


First differences of A116084.


2



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



OFFSET

1,4


COMMENTS

a(n1) 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..40.


FORMULA

a(n) = A116084(n+1)  A116084(n);
for primes p: a(p1) = A000009(p)  1.


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.
Sequence in context: A079501 A093335 A093333 * A329692 A216198 A085570
Adjacent sequences: A116082 A116083 A116084 * A116086 A116087 A116088


KEYWORD

nonn


AUTHOR

Reinhard Zumkeller, Feb 04 2006


EXTENSIONS

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


STATUS

approved



