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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
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
FORMULA
a(n) = A116084(n+1) - A116084(n);
for primes p: a(p-1) = 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 A369708
KEYWORD
nonn
AUTHOR
Reinhard Zumkeller, Feb 04 2006
EXTENSIONS
a(23)-a(40) from Giovanni Resta, Jul 15 2016
STATUS
approved