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 A263008 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)). 3
 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, 5, 31, 15, 1, 1, 11, 193, 3, 103, 3, 1, 8807, 1, 3383, 3, 21, 3, 8005, 1, 1, 13, 17, 3, 2047 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS 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). 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. 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. LINKS J. H. Conway, On Happy Factorizations, J. Integer Sequences, Vol. 1, 1998, #1. Wolfdieter Lang, Proof of a Theorem Related to the Happy Number Factorization. FORMULA 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)). EXAMPLE 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)). CROSSREFS Cf. A007970, A191856, A191857, A191860, A263009, A262026, A262027, A262028. Sequence in context: A019232 A185697 A321697 * A016479 A248843 A170910 Adjacent sequences:  A263005 A263006 A263007 * A263009 A263010 A263011 KEYWORD nonn AUTHOR Wolfdieter Lang, Oct 29 2015 STATUS approved

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Last modified November 18 17:49 EST 2018. Contains 317323 sequences. (Running on oeis4.)