A081233 -id:A081233 - cp's OEIS frontend

A002350 Take solution to Pellian equation x^2 - n*y^2 = 1 with smallest positive y and x >= 0; sequence gives a(n) = x, or 1 if n is a square. A002349 gives values of y.

1, 3, 2, 1, 9, 5, 8, 3, 1, 19, 10, 7, 649, 15, 4, 1, 33, 17, 170, 9, 55, 197, 24, 5, 1, 51, 26, 127, 9801, 11, 1520, 17, 23, 35, 6, 1, 73, 37, 25, 19, 2049, 13, 3482, 199, 161, 24335, 48, 7, 1, 99, 50, 649, 66249, 485, 89, 15, 151, 19603, 530, 31, 1766319049, 63, 8, 1

Offset: 1

N. J. A. Sloane

From A.H.M. Smeets, Nov 20 2017: (Start)
a(p*q^2) = b(p,q/gcd(A002349(p),q)) where
b(p,0) = 1, b(p,1) = a(p), b(p,i) = 2*a(p)*b(p,i-1) - b(p,i-2) for i>1. (End)

			For n = 1, 2, 3, 4, 5 solutions are (x,y) = (1, 0), (3, 2), (2, 1), (1, 0), (9, 4).

A. Cayley, Report of a committee appointed for the purpose of carrying on the tables connected with the Pellian equation ..., Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 13, pp. 430-443.
C. F. Degen, Canon Pellianus. Hafniae, Copenhagen, 1817.
D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 55.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
A. Cayley, Report of a committee appointed for the purpose of carrying on the tables connected with the Pellian equation ..., Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 13, pp. 430-443. (Annotated scanned copy)
L. Beeckmans, Squares expressible as sum of consecutive squares, Amer. Math. Monthly, 101 (1994), 437-442.
L. Euler, De solutione problematum diophanteorum per numeros integros (English and Latin), par. 17.
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A002349, A006702, A006703, A006704, A006705. See A033316, A033315, A033319 for records.

PellSolve[(m_Integer)?Positive] := Module[{cf, n, s}, cf = ContinuedFraction[ Sqrt[m]]; n = Length[ Last[cf]]; If[ OddQ[n], n = 2*n]; s = FromContinuedFraction[ ContinuedFraction[ Sqrt[m], n]]; {Numerator[s], Denominator[s]}]; f[n_] := If[ !IntegerQ[ Sqrt[n]], PellSolve[n][[1]], 1]; Table[ f[n], {n, 0, 65}]
Table[If[! IntegerQ[Sqrt[k]], {k, FindInstance[x^2 - k*y^2 == 1 && x > 0 && y > 0, {x, y}, Integers]}, Nothing], {k, 2, 80}][[All, 2, 1, 1, 2]] (* Horst H. Manninger, Mar 23 2021 *)

from sympy.ntheory.primetest import is_square
from sympy.solvers.diophantine.diophantine import diop_DN
def A002350(n): return 1 if is_square(n) else next(a for a,b in diop_DN(n,1)) # Chai Wah Wu, Feb 11 2025

a(prime(i)) = A081233(i). - R. J. Mathar, Feb 25 2025

A082393 Let p = n-th prime of the form 4k+1, take the integer solution (x,y) to the Pellian equation x^2 - p*y^2 = 1 with the smallest y >= 1; sequence gives value of y.

Original entry on oeis.org

4, 180, 8, 1820, 12, 320, 9100, 226153980, 267000, 53000, 6377352, 20, 15140424455100, 113296, 519712, 2113761020, 3726964292220, 190060, 183567298683461940, 448036604040, 28, 386460, 70255304, 649641205044600

Offset: 1

Cino Hilliard, Apr 14 2003

			For n = 1, p = 5, x=9, y=4 since 9^2 = 5*4^2 + 1, so a(1) = 4.

C. Stanley Ogilvy, Tomorrow's Math, 1972, p. 119.

Vincenzo Librandi, Table of n, a(n) for n = 1..2000
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Values of x are in A081232. Cf. A082394, A081233, A081234. Equals A002349(p).

PellSolve[(m_Integer)?Positive] := Module[{cf, n, s}, cf = ContinuedFraction[ Sqrt[m]]; n = Length[ Last[cf]]; If[ OddQ[n], n = 2*n]; s = FromContinuedFraction[ ContinuedFraction[ Sqrt[m], n]]; {Numerator[s], Denominator[s]}]; t = {}; Last /@ PellSolve /@ Select[Prime@Range@54, Mod[ #, 4] == 1 &] (* Robert G. Wilson v, Feb 28 2006 *)

p4xp1(n,m) = { forstep(p=1,m,4, for(y=1,n, if(isprime(p), x=y*y*p+1; if(issquare(x), print1(y" "); break; ) ) ) ) }

More terms from Robert G. Wilson v, Feb 28 2006

A081230 a(n) is the Levenshtein distance between n and n^n (where each is treated as a string).

Original entry on oeis.org

0, 1, 2, 3, 3, 4, 6, 8, 8, 9, 10, 11, 13, 16, 17, 18, 19, 22, 23, 26, 26, 28, 30, 32, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 86, 88, 90, 92, 94, 96, 99, 101, 103, 105, 107, 110, 112, 114, 116, 119, 121, 123, 125

Offset: 1

Francois Jooste (pin(AT)myway.com), Mar 11 2003

			a(9)=8 since we can transform 9 into 9^9=387420489 by 8 insertions, namely inserting 3,8,7,4,2,0,4 and 8 in front of 9. a(2)=1 since we can transform 2 into 2^2=4 by one substitution, namely 4 for 2.

M. Gilleland, Levenshtein Distance. [It has been suggested that this algorithm gives incorrect results sometimes. - _N. J. A. Sloane_]

Cf. A081231, A081232, A081233, A081234.

levenshtein[s_List, t_List] := Module[{d, n = Length@s, m = Length@t}, Which[s === t, 0, n == 0, m, m == 0, n, s != t, d = Table[0, {m + 1}, {n + 1}]; d[[1, Range[n + 1]]] = Range[0, n]; d[[Range[m + 1], 1]] = Range[0, m]; Do[ d[[j + 1, i + 1]] = Min[d[[j, i + 1]] + 1, d[[j + 1, i]] + 1, d[[j, i]] + If[ s[[i]] === t[[j]], 0, 1]], {j, m}, {i, n}]; d[[ -1, -1]] ]];
f[n_] := levenshtein[IntegerDigits[n], IntegerDigits[n^n]]; Array[f, 69] (* Robert G. Wilson v *)

Corrected by Robert G. Wilson v, Jan 25 2006

A081234 Let p = n-th prime, take smallest solution (x,y) to the Pellian equation x^2 - p*y^2 = 1 with x and y >= 1; sequence gives value of y.

Original entry on oeis.org

2, 1, 4, 3, 3, 180, 8, 39, 5, 1820, 273, 12, 320, 531, 7, 9100, 69, 226153980, 5967, 413, 267000, 9, 9, 53000, 6377352, 20, 22419, 93, 15140424455100, 113296, 419775, 927, 519712, 6578829, 2113761020, 140634693, 3726964292220, 5019135, 13, 190060

Offset: 1

N. J. A. Sloane, Apr 18 2003

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Values of x are in A081233. Equals A002349(p). Cf. A082393.

PellSolve[(m_Integer)?Positive] := Module[{cf, n, s}, cf = ContinuedFraction[ Sqrt[m]]; n = Length[Last[cf]]; If[OddQ[n], n = 2*n]; s = FromContinuedFraction[ ContinuedFraction[ Sqrt[m], n]]; {Numerator[s], Denominator[s]}]; Table[ PellSolve[ Prime[n]][[2]], {n, 40}] (* Robert G. Wilson v, Jul 22 2005 *)

More terms (a(8) - a(40)) from Robert G. Wilson v, Jul 22 2005

A081232 Let p = n-th prime of the form 4k+1, take smallest solution (x,y) to the Pellian equation x^2 - p*y^2 = 1 with x and y >= 1; sequence gives value of x.

Original entry on oeis.org

9, 649, 33, 9801, 73, 2049, 66249, 1766319049, 2281249, 500001, 62809633, 201, 158070671986249, 1204353, 6083073, 25801741449, 46698728731849, 2499849, 2469645423824185801, 6224323426849, 393, 5848201, 1072400673

Offset: 1

N. J. A. Sloane, Apr 18 2003

			For n = 1, p = 5, x=9, y=4 since 9^2 = 5*4^2 + 1, so a(1) = 9.

Vincenzo Librandi, Table of n, a(n) for n = 1..2000
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Values of y are in A082393. Cf. A082394, A081233. Equals A002350(p).

PellSolve[(m_Integer)?Positive] := Module[{cof, n, s}, cof = ContinuedFraction[Sqrt[m]]; n = Length[Last[cof]]; If[ OddQ[n], n = 2*n]; s = FromContinuedFraction[ContinuedFraction[Sqrt[m], n]]; {Numerator[s], Denominator[s]}]; First /@ PellSolve /@ Select[Prime@Range@54, Mod[ #, 4] == 1 &] (* Robert G. Wilson v *)

More terms from Robert G. Wilson v, Feb 28 2006

A173202 Solutions y of the Mordell equation y^2 = x^3 - 3a^2 + 1 for a = 0,1,2, ... (solutions x are given by the sequence A000466).

Original entry on oeis.org

0, 5, 58, 207, 500, 985, 1710, 2723, 4072, 5805, 7970, 10615, 13788, 17537, 21910, 26955, 32720, 39253, 46602, 54815, 63940, 74025, 85118, 97267, 110520, 124925, 140530, 157383, 175532, 195025, 215910, 238235, 262048, 287397, 314330, 342895

Offset: 1

Michel Lagneau, Feb 12 2010

For many values of k for the equation y^2 = x^3 + k, all the solutions are known. For example, we have solutions for k=-2: (x,y) = (3,-5) and (3,5). A complete resolution for all integers k is unknown. Theorem: Let k be < -1, free of square factors, with k == 2 or 3 (mod 4). Suppose that the number of classes h(Q(sqrt(k))) is not divisible by 3. Then the equation y^2 = x^3 + k admits integer solutions if and only if k = 1 - 3a^2 or 1 - 3a^2 where a is an integer. In this case, the solutions are x = a^2 - k, y = a(a^2 + 3k) or -a(a^2 + 3k) (the first reference gives the proof of this theorem). With k = -1 - 3a^2, we obtain the solutions x = 4a^2 + 1, y = a(8a^2 + 3) or -a(8a^2 + 3). For the case k = 1 - 3a^2, we obtain the solution x = 4a^2 - 1 given by the sequence A000466.

			With a=3, x = 35 and y = 207, and then 207^2 = 35^2 - 26.

T. Apostol, Introduction to Analytic Number Theory, Springer, 1976
D. Duverney, Theorie des nombres (2e edition), Dunod, 2007, p.151

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
W. J. Ellison, F. Ellison, J. Pesek, C. E. Stall & D. S. Stall, The diophantine equation y^2 + k = x^3, J. Number Theory 4 (1972), 107-117.
Helmut Richter, Solutions of Mordell's equation y^2 = x^3 + k (solutions for 0
School of Mathematics and Statistics, University of St Andrews, Louis Joel Mordell.
Eric Weisstein's World of Mathematics, Mordell Curve.
D. J. Wright, Mordell's Equation.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Diophantine equations: see also Pellian equation: (A081233, A081234), (A081231, A082394), (A081232, A082393); Mordell equation: A053755, A173200; Diophantine equations: A006452, A006451, A006454.

I:=[0, 5, 58, 207]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Jul 02 2012

for a from 0 to 100 do : z := evalf(a*(8*a^2 - 3)) : print (z) :od :

CoefficientList[Series[x*(5+38*x+5*x^2)/(1-x)^4,{x,0,40}],x] (* Vincenzo Librandi, Jul 02 2012 *)
CoefficientList[Series[E^x (5 x + 24 x^2 + 8 x^3), {x, 0, 40}], x]*Table[n!, {n, 0, 40}] (* Stefano Spezia, Dec 04 2018 *)

y = a*(8*a^2 - 3).
a(n) = sqrt(A000466(n)^3 - A080663(n)). - Artur Jasinski, Nov 26 2011
From Colin Barker, Apr 26 2012: (Start)
a(n) = 8*n^3 - 24*n^2 + 21*n - 5.
G.f.: x^2*(5 + 38*x + 5*x^2)/(1 - x)^4. (End)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Vincenzo Librandi, Jul 02 2012
E.g.f.: exp(x)*(5*x + 24*x^2 + 8*x^3). - Stefano Spezia, Dec 04 2018

cp's OEIS Frontend

A002350 Take solution to Pellian equation x^2 - n*y^2 = 1 with smallest positive y and x >= 0; sequence gives a(n) = x, or 1 if n is a square. A002349 gives values of y.

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A082393 Let p = n-th prime of the form 4k+1, take the integer solution (x,y) to the Pellian equation x^2 - p*y^2 = 1 with the smallest y >= 1; sequence gives value of y.

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A081230 a(n) is the Levenshtein distance between n and n^n (where each is treated as a string).

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A081234 Let p = n-th prime, take smallest solution (x,y) to the Pellian equation x^2 - p*y^2 = 1 with x and y >= 1; sequence gives value of y.

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A081232 Let p = n-th prime of the form 4k+1, take smallest solution (x,y) to the Pellian equation x^2 - p*y^2 = 1 with x and y >= 1; sequence gives value of x.

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A173202 Solutions y of the Mordell equation y^2 = x^3 - 3a^2 + 1 for a = 0,1,2, ... (solutions x are given by the sequence A000466).

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