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A227127 The Akiyama-Tanigawa algorithm applied to 1/(1,2,3,5,... old prime numbers). Reduced numerators of the second row.

%I A227127 #16 Jul 09 2013 06:59:28
%S A227127 1,1,2,8,20,12,28,16,36,60,22,72,52,28,60,96,102,36,114,80,42,132,92,
%T A227127 144,200,104,54,112,58,120,434,128,198,68,350,72,222,228,156,240,246,
%U A227127 84,430,88,180,92,564,576,196,100,204,312,106,540,330,336,342,116,354,240,122
%N A227127 The Akiyama-Tanigawa algorithm applied to 1/(1,2,3,5,... old prime numbers). Reduced numerators of the second row.
%C A227127 1/A008578(n) and successive rows:
%C A227127 1,          1/2,     1/3,     1/5,   1/7,
%C A227127 1/2,        1/3,     2/5,    8/35,           = c(n) = a(n)/b(n)
%C A227127 1/6,      -2/15,   18/35,
%C A227127 3/10,  -136/105,
%C A227127 67/42
%C A227127 b(n) is essentially A006094. See A209329.
%C A227127 a(n) yields to a permutation of A008578 (via 1, 1, 2, 8, 12, 16, 20, 22, ...): 1, 2, 3, 5, 11, 17, 7, 29,... .
%F A227127 a(n) = (n+1)*A001223(n-1), for n>=3.
%e A227127 a(n) is the numerators of c(n): c(0) = 1-1/2 = 1/2, c(1) = 2*(1/2-1/3) = 1/3, c(2) = 3*(1/3-1/5) = 2/5, c(3) = 4*(1/5-1/7)=8/35.
%e A227127 a(3) = 4*2 = 8, a(4) = 5*4 = 20.
%t A227127 a[0, 0] = 1; a[0, m_ /; m > 0] := 1/Prime[m]; a[n_, m_] := a[n, m] = (m+1)*(a[n-1, m ] - a[n-1, m+1]); Table[a[1, m] // Numerator, {m, 0, 60}] (* _Jean-François Alcover_, Jul 04 2013 *)
%Y A227127 Cf. A002110, A006954.
%K A227127 nonn,frac
%O A227127 0,3
%A A227127 _Paul Curtz_, Jul 02 2013