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A206350 Position of 1/n in the canonical bijection from the positive integers to the positive rational numbers.

%I A206350 #38 Aug 02 2023 07:13:13
%S A206350 1,2,4,8,12,20,24,36,44,56,64,84,92,116,128,144,160,192,204,240,256,
%T A206350 280,300,344,360,400,424,460,484,540,556,616,648,688,720,768,792,864,
%U A206350 900,948,980,1060,1084,1168,1208,1256,1300,1392,1424,1508,1548
%N A206350 Position of 1/n in the canonical bijection from the positive integers to the positive rational numbers.
%C A206350 The canonical bijection from the positive integers to the positive rational numbers is given by A038568(n)/A038569(n).
%C A206350 Appears to be a variant of A049696. - _R. J. Mathar_, Feb 11 2012
%C A206350 Apparently numbers m such that A071912(m) = 1. - _Bill McEachen_, Aug 01 2023
%H A206350 G. C. Greubel, <a href="/A206350/b206350.txt">Table of n, a(n) for n = 1..10000</a>
%F A206350 a(1) = 1, a(n+1) = Sum_{k=1..n} mu(k) * floor(n/k) * floor(1 + n/k), where mu(k) is the Moebius function A008683. - _Daniel Suteu_, May 28 2018
%F A206350 a(n) = 2*Sum_{k=1..n-1} A000010(k), a(1) = 1. - _Robert Israel_, Apr 24 2015
%e A206350 The canonical bijection starts with 1/1, 1/2, 2/1, 1/3, 3/1, 2/3, 3/2, 1/4, 4/1, 3/4, 4/3, 1/5, 5/1, so that A206297 starts with 1,3,5,9,13 and this sequence starts with 1,2,4,8,12.
%p A206350 1, op(2*ListTools:-PartialSums(map(numtheory:-phi, [$1..100]))); # _Robert Israel_, Apr 24 2015
%t A206350 a[n_]:= Module[{s=1, k=2, j=1},
%t A206350   While[s<=n, s= s + 2*EulerPhi[k]; k= k+1];
%t A206350   s = s - 2*EulerPhi[k-1];
%t A206350   While[s<=n, If[GCD[j, k-1] == 1,
%t A206350     s = s+2]; j = j+1];
%t A206350   If[s>n+1, j-1, k-1]];
%t A206350 t = Table[a[n], {n, 0, 3000}];   (* A038568 *)
%t A206350 ReplacePart[Flatten[Position[t, 1]], 1, 1] (* A206350 *)
%t A206350 (* Second program *)
%t A206350 a[n_]:= If[n==1, 1, 2*Sum[EulerPhi[k], {k, n-1}]];;
%t A206350 Table[a[n], {n, 80}] (* _G. C. Greubel_, Mar 29 2023 *)
%o A206350 (Magma) [1] cat [2*(&+[EulerPhi(k): k in [1..n-1]]): n in [2..80]]; // _G. C. Greubel_, Mar 29 2023
%o A206350 (SageMath)
%o A206350 def A206350(n): return 1 if (n==1) else 2*sum(euler_phi(k) for k in range(1,n))
%o A206350 [A206350(n) for n in range(1,80)] # _G. C. Greubel_, Mar 29 2023
%Y A206350 Cf. A000010, A008683, A038568, A038569, A049696, A171503, A206296.
%Y A206350 Cf. A071912.
%K A206350 nonn
%O A206350 1,2
%A A206350 _Clark Kimberling_, Feb 06 2012