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A116085 First differences of A116084.

%I A116085 #17 Dec 14 2024 04:31:52
%S A116085 0,1,1,2,2,4,5,8,13,11,23,17,45,151,151,37,301,53,1009,2534,1177,103,
%T A116085 4275,6541,3479,12380,43589,255,64634,339,97373,299183,60599,1957769,
%U A116085 2118020,759,310542,4731201,14267125,1259,24538635,1609,57443858,260450002,8940128
%N A116085 First differences of A116084.
%C A116085 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
%F A116085 a(n) = A116084(n+1) - A116084(n).
%F A116085 a(p-1) = A000009(p) - 1 for prime p.
%e A116085 a(1) = 0 since there is no way to write 1 as sum of distinct fractions with denominator not larger than 2.
%e A116085 a(2) = # [1/3+2/3] = 1,
%e A116085 a(3) = # [1/4+3/4] = 1,
%e A116085 a(4) = # [1/5+4/5, 2/5+3/5] = 2,
%e A116085 a(5) = # [1/6+5/6, 1/6+1/3+1/2] = 2.
%t A116085 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 *)
%Y A116085 Cf. A115856, A116084, A154888.
%K A116085 nonn
%O A116085 1,4
%A A116085 _Reinhard Zumkeller_, Feb 04 2006
%E A116085 a(23)-a(40) from _Giovanni Resta_, Jul 15 2016
%E A116085 More terms from _Jinyuan Wang_, Dec 14 2024