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%I #11 Oct 31 2015 14:34:50
%S 1,1,1,3,1,13,1,1,5,7,1,1,3,59,1,1,7,23,1,221,7,1,1,1,9,3,7,11,1,1,47,
%T 5,31,15,1,1,11,193,3,103,3,1,8807,1,3383,3,21,3,8005,1,1,13,17,3,2047
%N First member T0(n) of the smallest positive pair (T0(n), U0(n)) for the n-th 2-happy number couple (D(n), E(n)).
%C The 2-happy numbers D(n)*E(n) are given in A007970(n) (called rhombic numbers in the Conway paper). D(n) = A191856(n), E(n) = A191857(n). Here the corresponding smallest positive numbers satisfying E(n)*U(n)^2 - D(n)*T(n)^2 = +2, n >= 1, with odd U(n) and T(n) are given as T0(n) = a(n) and U0(n) = A263009(n).
%C In the W. Lang link the first U0(n) and T0(n) numbers are given in the Table for d(n) = A007970(n), n >= 1.
%C In the Zumkeller link "Initial Happy Factorization Data" given in A191860 the a(n) = T0(n) numbers appear for the t = 2 rows in column v.
%H 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.
%H Wolfdieter Lang, <a href="/A007970/a007970.pdf">Proof of a Theorem Related to the Happy Number Factorization.</a>
%F A191857(n)*A263009(n)^2 - A191856(n)*a(n)^2 = +2, and a(n) with A263009(n) is the smallest positive solution for the given 2-happy couple (A191856(n), A191857(n)).
%e n = 6: 2-happy number A007970(6) = 19 = 1*19 = A191856(6)*A191857(6). 19*A263009(6)^2 - 1*a(6)^2 = 19*3^2 - 1*13^2 = +2. This is the smallest positive solution for the given 2-happy couple (A191856(n), A191857(n)).
%Y Cf. A007970, A191856, A191857, A191860, A263009, A262026, A262027, A262028.
%K nonn
%O 1,4
%A _Wolfdieter Lang_, Oct 29 2015
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