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%I #13 Jul 16 2016 07:26:43
%S 0,1,1,2,2,4,5,8,13,11,23,17,45,151,151,37,301,53,1009,2534,1177,103,
%T 4275,6541,3479,12380,43589,255,64634,339,97373,299183,60599,1957769,
%U 2118020,759,310542,4731201,14267125,1259
%N First differences of A116084.
%C 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 a(n) = A116084(n+1) - A116084(n);
%F for primes p: a(p-1) = A000009(p) - 1.
%e a(1) = 0 since there is no way to write 1 as sum of distinct fractions with denominator not larger than 2.
%e a(2) = # [1/3+2/3] = 1,
%e a(3) = # [1/4+3/4] = 1,
%e a(4) = # [1/5+4/5, 2/5+3/5] = 2,
%e a(5) = # [1/6+5/6, 1/6+1/3+1/2] = 2.
%t 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 Cf. A115856.
%K nonn
%O 1,4
%A _Reinhard Zumkeller_, Feb 04 2006
%E a(23)-a(40) from _Giovanni Resta_, Jul 15 2016
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