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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)). 2
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, 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, 151, 1, 5604, 1, 10, 5, 2, 4005, 5, 8890182, 1, 7, 3, 776, 16, 35, 6, 277 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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).

For a proof of Conway's happy number factorization theorem see the W. Lang link under A007970.

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.

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.

LINKS

Table of n, a(n) for n=1..74.

J. H. Conway, On Happy Factorizations, J. Integer Sequences, Vol. 1, 1998, #1.

FORMULA

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)).

EXAMPLE

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).

CROSSREFS

Cf. A007969, A007970, A191854, A191855, A191860, A263007, A262025, A261250, A263008, A263009.

Sequence in context: A057082 A218951 A096642 * A261250 A305354 A227860

Adjacent sequences:  A263003 A263004 A263005 * A263007 A263008 A263009

KEYWORD

nonn

AUTHOR

Wolfdieter Lang, Oct 28 2015

STATUS

approved

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Last modified October 20 15:22 EDT 2018. Contains 316388 sequences. (Running on oeis4.)