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A154888 Number of ways to partition 1 into distinct reduced fractions i/j with j<=n. 4

%I #15 Oct 19 2017 10:38:29

%S 1,1,2,3,5,7,11,16,24,37,48,71,88,133,284,435,472,773,826,1835,4369,

%T 5546,5649,9924,16465,19944,32324,75913,76168,140802,141141,238514,

%U 537697,598296,2556065,4674085,4674844,4985386,9716587,23983712,23984971

%N Number of ways to partition 1 into distinct reduced fractions i/j with j<=n.

%C a(n) = A116084(n)+1 for all n because the decompositions are the same except for the additional fraction 1/1 allowed here but excluded in A116084. - _M. F. Hasler_, Jul 14 2016

%e a(6) = #[1, 5/6+1/6, 4/5+1/5, 3/4+1/4, 2/3+1/3, 3/5+2/5, 1/2+1/3+1/6] = 7.

%t Farey[n_] := Union@ Flatten@ Table[ a/b, {b, n}, {a, b}]; t[n_, k_] := t[n, k] = Block[{c = j = 0, ip = IntegerPartitions[1, {k}, Farey@ n]}, len = 1 + Length@ ip; While[j < len, If[Plus @@ Union@ ip[[j]] == 1, c++ ]; j++ ]; c]; f[n_] := Plus @@ Table[ t[n, k], {k, Ceiling[n/2]}]; Array[f, 24] (* _Robert G. Wilson v_, Aug 30 2010 *)

%Y Cf. A119983, A154887.

%Y Equals A116084(n) + 1.

%K more,nonn

%O 1,3

%A _Reinhard Zumkeller_, Jan 18 2009

%E a(22)-a(26) from _Robert G. Wilson v_, Aug 30 2010

%E a(27)-a(34) from _Don Reble_, Jul 13 2016

%E a(35)-a(41) from _Giovanni Resta_, Jul 15 2016

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Last modified March 28 17:25 EDT 2024. Contains 371254 sequences. (Running on oeis4.)