cp's OEIS Frontend

A263006 First member R0(n) of the smallest positive pair (R0(n), S0(n)) for the n-th 1-happy number couple (B(n), C(n)).

%I A263006 #17 Oct 31 2015 14:33:04
%S A263006 1,2,1,3,1,18,1,4,2,1,3,7,5,3,70,1,1,1,6,3,2,32,1,3,4,23,7,9,182,11,2,
%T A263006 1,5,99,1,29718,1,8,4,2,13,5,1,1068,43,39,5,1,9,3,378,51,500,1,5,45,
%U A263006 151,1,5604,1,10,5,2,4005,5,8890182,1,7,3,776,16,35,6,277
%N A263006 First member R0(n) of the smallest positive pair (R0(n), S0(n)) for the n-th 1-happy number couple (B(n), C(n)).
%C A263006 The 1-happy numbers B(n)*C(n) are given in A007969(n) (called rectangular numbers in the Conway paper). B(n) = A191854(n), C(n) = A191855(n). Here the corresponding smallest positive numbers satisfying C(n)*S0(n)^2 - B(n)*R0(n)^2 = +1, n >= 1, are given as R0(n) = a(n) and S0(n) = A263007(n).
%C A263006 For a proof of Conway's happy number factorization theorem see the W. Lang link under A007970.
%C A263006 In the W. Lang link given in A007969 the first C(n), B(n), S0(n), R0(n) numbers are given in the Table for d(n) = A007969(n), n >= 1.
%C A263006 In the Zumkeller link "Initial Happy Factorization Data" given in A191860 the a(n) = R0(n) numbers appear for the t = 1 rows in column v.
%H A263006 J. H. Conway, <a href="http://www.cs.uwaterloo.ca/journals/JIS/happy.html">On Happy Factorizations</a>, J. Integer Sequences, Vol. 1, 1998, #1.
%F A263006 A191855(n)*A263007(n)^2 - A191854(n)*a(n)^2  = +1, and a(n) with A263007(n) is the smallest positive solution for the given 1-happy couple (A191854(n), A191855(n)).
%e A263006 n = 6: 1-happy number A007969(6) = 13 = 1*13 = A191854(6)*A191855(6). 13*A263007(6)^2 - 1*a(6)^2 = 13*5^2 - 1*18^2 = +1. This is the smallest positive solution for (B, C) = (1, 13).
%Y A263006 Cf. A007969, A007970, A191854, A191855, A191860, A263007, A262025, A261250, A263008, A263009.
%K A263006 nonn
%O A263006 1,2
%A A263006 _Wolfdieter Lang_, Oct 28 2015