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 A298211 Smallest n such that A001353(a(n)) == 0 (mod n), i.e., x=A001075(a(n)) and y=A001353(a(n)) is the fundamental solution of the Pell equation x^2 - 3*(n*y)^2 = 1. 4
 1, 2, 3, 2, 3, 6, 4, 4, 9, 6, 5, 6, 6, 4, 3, 8, 9, 18, 5, 6, 12, 10, 11, 12, 15, 6, 27, 4, 15, 6, 16, 16, 15, 18, 12, 18, 18, 10, 6, 12, 7, 12, 11, 10, 9, 22, 23, 24, 28, 30, 9, 6, 9, 54, 15, 4, 15, 30, 29, 6, 30, 16, 36, 32, 6, 30, 17, 18, 33, 12, 7, 36, 18, 18, 15 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS The fundamental solution of the Pell equation x^2 - 3*(n*y)^2 = 1 is the smallest solution of x^2 - 3*y^2 = 1 satisfying y == 0 (mod n). REFERENCES Michael J. Jacobson, Jr. and Hugh C. Williams, Solving the Pell Equation, Springer, 2009, pages 1-17. LINKS A.H.M. Smeets, Table of n, a(n) for n = 1..20000 H. W. Lenstra Jr., Solving the Pell Equation, Notices of the AMS, Vol.49, No.2, Feb. 2002, p.182-192. FORMULA a(n) <= n. a(A038754(n)) = A038754(n). A001075(a(n)) = A002350(3*n^2). A001353(a(n)) = A002349(3*n^2). if n | m then a(n) | a(m). a(3^m) = 3^m  and a(2*3^m) = 2*3^m for m>=0. In general: if p is prime and p == 3 (mod 4) then: a(n) = n iff n = p^m or n = 2*p^m, for m>=0. a(k*A005385(n)) = a(k)*A005384(n) for n>2 and k > 0 (conjectured). a(p) | (p-A091338(p)) for p is an odd prime. - A.H.M. Smeets, Aug 02 2018 MATHEMATICA With[{s = Array[ChebyshevU[-1 + #, 2] &, 75]}, Table[FirstPosition[s, k_ /; Divisible[k, n]][[1]], {n, Length@ s}]] (* Michael De Vlieger, Jan 15 2018, after Eric W. Weisstein at A001353 *) PROG Python: xf, yf = 2, 1 x, n = 2*xf, 0 while n < 20000: ....n = n+1 ....y1, y0, i = 0, yf, 1 ....while y0%n != 0: ........y1, y0, i = y0, x*y0-y1, i+1 ....print(n, i) CROSSREFS Cf. A091338, A298210, A298212. Sequence in context: A278910 A235669 A118088 * A088212 A085208 A257302 Adjacent sequences:  A298208 A298209 A298210 * A298212 A298213 A298214 KEYWORD nonn AUTHOR A.H.M. Smeets, Jan 15 2018 STATUS approved

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Last modified December 14 22:42 EST 2019. Contains 329987 sequences. (Running on oeis4.)