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A077233
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a(n) is smallest natural number satisfying Pell equation b^2- d(n)*a^2= +1 or = -1, with d(n)=A000037(n) (nonsquare). Corresponding smallest b(n)=A077232(n).
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5
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1, 1, 1, 2, 3, 1, 1, 3, 2, 5, 4, 1, 1, 4, 39, 2, 12, 42, 5, 1, 1, 5, 24, 13, 2, 273, 3, 4, 6, 1, 1, 6, 4, 3, 5, 2, 531, 30, 24, 3588, 7, 1, 1, 7, 90, 25, 66, 12, 2, 20, 13, 69, 4, 3805, 8, 1, 1, 8, 5967, 4, 936, 30, 413, 2, 125, 5, 3, 6630, 40, 6, 9, 1, 1, 9, 6, 41, 1122, 3, 21, 53, 2, 165, 120, 1260, 221064, 4, 5, 569, 10, 1, 1, 10, 22419
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OFFSET
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1,4
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COMMENTS
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If d(n)=A000037(n) is from A003654 (that is if the regular continued fraction for sqrt(d(n)) has odd (primitive) period length) then the -1 option applies. For such d(n) the minimal b(n) and a(n) numbers for the +1 option are 2*b(n)^2 + 1 and 2*b(n)*a(n), respectively (see Perron I, pp. 94,p5).
For general integer solutions see A077232 comments.
If the trivial solution x=1, y=0 is included, the sequence becomes A006703. - T. D. Noe, May 17 2007
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REFERENCES
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T. Nagell, "Introduction to Number Theory", Chelsea Pub., New York, 1964, table p. 301.
O. Perron, "Die Lehre von den Kettenbruechen, Bd.I", Teubner, 1954, 1957 (Sec. 26, p. 91 with explanation on pp. 94,95).
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LINKS
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Ray Chandler, Table of n, a(n) for n = 1..10000
A. M. Legendre, Fractions les plus simples m/n qui satisfont à l'équation m^2 - an^2 =+-1 pour tout nombre non quarré a depuis 2 jusqu'à 1003, Essai sur la Théorie des Nombres An VI, Table XII. [Paul Curtz, Apr 10 2019]
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FORMULA
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a(n)=sqrt((A077232(n)^2 - (-1)^(c(n)))/A000037(n)) with c(n)=1 if A000037(n)=A003654(k) for some k>=1 else c(n)=0.
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EXAMPLE
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d=10=A000037(7)=A003654(3), therefore a(7)=1 and b(7)=A077232(7)=3 give 3^2=10*1^2 -1 and 2*b(7)^2+1=19 and 2*b(7)*a(7)=2*3*1=6 satisfy 19^2 - 10*6^2 = +1.
d=11=A000037(8) is not in A003654, therefore there is no (nontrivial) solution of the b^2 - d*a^2 = -1 Pell equation and a(8)=3 and b(8)=A077232(8)=10 satisfy 10^2 - 11*3^2 = +1. See A077232 for further examples.
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MATHEMATICA
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d[n_] := d[n] = n + Floor[Sqrt[n] + 1/2]; r[n_, a_] := Reduce[lhs = b^2 - d[n]*a^2; b > 0 && (lhs == 1 || lhs == -1) , {b}, Integers]; r[n_] := For[a = 1, True, a++, If[r[n, a] =!= False, Return[a]]]; A077233 = Table[a = r[n]; Print["a(", n, ") = ", a]; a, {n, 1, 93}] (* Jean-François Alcover, May 22 2012 *)
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CROSSREFS
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Cf. A000037, A003654, A003814, A033317, A077232.
Sequence in context: A296659 A270823 A067627 * A282290 A178795 A123185
Adjacent sequences: A077230 A077231 A077232 * A077234 A077235 A077236
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KEYWORD
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nonn,nice
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AUTHOR
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Wolfdieter Lang, Nov 08 2002
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STATUS
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approved
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