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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.)