A094048 -id:A094048 - cp's OEIS frontend

A094049 Let p(n) be the n-th prime congruent to 1 mod 4. Then a(n) = the least k for which m^2+1=p(n)*k^2 has a solution.

1, 5, 1, 13, 1, 5, 25, 3805, 125, 53, 569, 1, 851525, 73, 149, 9305, 385645, 85, 82596761, 126985, 1, 113, 1517, 4574225, 1, 5, 535979945, 63445, 145, 7170685, 19805, 55335641, 493, 3793, 265, 65, 1027776565, 1

Offset: 1

Matthijs Coster, Apr 29 2004

nonn

Cf. A002144, A094048.

 f[n_] := Block[{y = 1}, While[ !IntegerQ[ Sqrt[n*y^2 - 1]], y++]; y]; lst = {}; Do[p = Prime@n; If[Mod[p, 4] == 1, AppendTo[lst, f@p]; Print[{n, f@p}]], {n, 66}]; lst

Edited by Don Reble, Apr 30 2004

A191860 First member of a pair of numbers occurring in the definition of 1-happy couples.

Original entry on oeis.org

1, 2, 2, 3, 3, 1, 18, 4, 4, 13, 1, 3, 5, 5, 3, 70, 1, 1, 6, 6, 3, 3, 32, 59, 3, 4, 7, 7, 9, 182, 11, 2, 1, 5, 23, 1, 29718, 8, 8, 221, 2, 13, 7, 1, 1068, 1, 39, 5, 9, 9, 3, 378, 7, 500, 11, 5, 45, 151, 1, 5604, 10, 10, 5, 2, 31, 5, 8890182, 1, 7, 3, 776, 15

Offset: 1

Reinhard Zumkeller, Jun 18 2011

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..200
J. H. Conway, On Happy Factorizations, J. Integer Sequences, Vol. 1, 1998, #1.
Initial Happy Factorization Data

Cf. A007968, A007969, A094048 (subsequence).

a191860 = fst . sqrtPair . a007969
Function sqrtPair is defined in A007968.
-- Reinhard Zumkeller, Oct 11 2015

Wrong comment and wrong formula removed (thanks to Wolfdieter Lang, who pointed this out) by Reinhard Zumkeller, Oct 11 2015

A130226 Smallest integer x satisfying the Pell equation x^2-k*y^2=-1 for the values of k given in A031396.

Original entry on oeis.org

0, 1, 2, 3, 18, 4, 5, 70, 6, 32, 7, 182, 99, 29718, 8, 1068, 43, 9, 378, 500, 5604, 10, 4005, 8890182, 776, 11, 682, 57, 1744, 12, 113582, 4832118, 13, 1118, 1111225770, 68, 1764132, 14, 3141, 251, 15, 1710, 23156, 71011068, 4443, 16, 6072, 82, 1407

Offset: 1

Colin Barker, Aug 05 2007

nonn

			a(5)=18 because A031396(5)=13, and the solution to x^2-13y^2=-1 with smallest possible x has x=18.

Ray Chandler, Table of n, a(n) for n = 1..10000
K. Lakshmi and R. Someshwari, On The Negative Pell Equation y^2 = 72x^2 - 23, International Journal of Emerging Technologies in Engineering Research (IJETER), Volume 4, Issue 7, July (2016).
R. Suganya and D. Maheswari, On the Negative Pellian Equation y^2 = 110 * x^2 - 29, Journal of Mathematics and Informatics, Vol. 11 (2017), pp. 63-71.
S. Vidhyalakshmi, V. Krithika, and K. Agalya, On The Negative Pell Equation y^2 = 72x^2 - 8, International Journal of Emerging Technologies in Engineering Research (IJETER) Volume 4, Issue 2, February (2016).

Cf. A094048.

A130226 := proc(m)
    local xm,x ,i,xmo,y2;
    xm := [] ; # x^2-m*y^2=-1 (mod m) requires x in xm[]
    for x from 0 to m-1 do
        if modp(x^2,m) = modp(-1,m) then
            xm := [op(xm),x] ;
        end if;
    end do:
    for i from 0 do
        for xmo in xm do
            x := i*m+xmo ;
            y2 := (x^2+1)/m ;
            if issqr(y2) then
                return x ;
            end if;
        end do:
    end do:
end proc:
L := BFILETOLIST("b031396.txt") ;
n := 1:
for m in L do
    printf("%d %d\n",n,A130226(m)) ;
    n := n+1 ;
end do: # R. J. Mathar, Oct 19 2014

terms = 1000;
a031396 = Cases[Import["https://oeis.org/A031396/b031396.txt", "Table"], {, }][[;; terms, 2]];
sol[n_] := Solve[x > 0 && y > 0 && x^2 - n y^2 == -1, {x, y}, Integers];
a[1] = 0; a[n_] := a[n] = x /. sol[a031396[[n]]] /. C[1] -> 0 // First // Simplify // Quiet;
Table[Print[n, " ", a031396[[n]], " ", a[n]]; a[n], {n, 1, terms}] (* Jean-François Alcover, Apr 05 2020 *)

A306529 x-value of the smallest solution to x^2 - py^2 = 2(-1)^((p+1)/4), p = A002145(n).

Original entry on oeis.org

1, 3, 3, 13, 5, 39, 59, 7, 23, 221, 59, 9, 9, 477, 31, 2175, 103, 8807, 41571, 8005, 13, 2047, 2999, 127539, 527593, 15, 15, 2489, 1917, 373, 340551, 11759, 9409, 4109, 52778687, 801, 19, 137913, 113759383, 137, 16437, 12311, 21, 21, 15732537, 1275, 1729, 7204587, 305987, 67

Offset: 1

Jianing Song, Mar 25 2019

nonn

a(n) exists for all n.

X = a(n)^2 - (-1)^((p+1)/4), Y = a(n)*A306566(n) gives the smallest solution to x^2 - p*y^2 = 1, p = A002145(n). As a result, all the positive solutions to x^2 - p*y^2 = 2*(-1)^((p+1)/4) are given by (x_n, y_n) where x_n + (y_n)*sqrt(p) = (a(n) + A306566(n)*sqrt(p))*(X + Y*sqrt(p))^n.

			The smallest solution to x^2 - p*y^2 = 2*(-1)^((p+1)/4) for the first primes congruent to 3 modulo 4:
  n |      Equation     | x_min | y_min
  1 | x^2 -  3*y^2 = -2 |     1 |     1
  2 | x^2 -  7*y^2 = +2 |     3 |     1
  3 | x^2 - 11*y^2 = -2 |     3 |     1
  4 | x^2 - 19*y^2 = -2 |    13 |     3
  5 | x^2 - 23*y^2 = +2 |     5 |     1
  6 | x^2 - 31*y^2 = +2 |    39 |     7
  7 | x^2 - 43*y^2 = -2 |    59 |     9
  8 | x^2 - 47*y^2 = +2 |     7 |     1
  9 | x^2 - 59*y^2 = -2 |    23 |     3

Cf. A002145, A306566 (y-values).

Similar sequences: A094048, A094049 (x^2 - A002144(n)*y^2 = -1); A306618, A306619 (2*x^2 - A002145(n)*y^2 = (-1)^((p+1)/4)).

b(p) = if(isprime(p)&&p%4==3, x=1; while(!issquare((x^2 - 2*(-1)^((p+1)/4))/p), x++); x)
forprime(p=3, 500, if(p%4==3, print1(b(p), ", ")))

If the continued fraction of sqrt(A002145(n)) is [a_0; {a_1, a_2, ..., a_(k-1), a_k, a_(k-1), ..., a_1, 2*a_0}], where {} is the periodic part, let x/y = [a_0; a_1, a_2, ..., a_(k-1)], gcd(x, y) = 1, then a(n) = x and A306566(n) = y.

A306566 y-value of the smallest solution to x^2 - py^2 = 2(-1)^((p+1)/4), p = A002145(n).

Original entry on oeis.org

1, 1, 1, 3, 1, 7, 9, 1, 3, 27, 7, 1, 1, 47, 3, 193, 9, 747, 3383, 627, 1, 153, 217, 9041, 36321, 1, 1, 161, 121, 23, 20687, 699, 537, 233, 2900979, 43, 1, 7199, 5843427, 7, 803, 593, 1, 1, 731153, 59, 79, 326471, 13809, 3, 7, 12507, 541137, 11, 563210019

Offset: 1

Jianing Song, Mar 25 2019

nonn

a(n) exists for all n.

X = A306529(n)^2 - (-1)^((p+1)/4), Y = A306529(n)*a(n) gives the smallest solution to x^2 - p*y^2 = 1, p = A002145(n). As a result, all the positive solutions to x^2 - p*y^2 = 2*(-1)^((p+1)/4) are given by (x_n, y_n) where x_n + (y_n)*sqrt(p) = (A306529(n) + a(n)*sqrt(p))*(X + Y*sqrt(p))^n.

			The smallest solution to x^2 - p*y^2 = 2*(-1)^((p+1)/4) for the first primes congruent to 3 modulo 4:
  n |      Equation     | x_min | y_min
  1 | x^2 -  3*y^2 = -2 |     1 |     1
  2 | x^2 -  7*y^2 = +2 |     3 |     1
  3 | x^2 - 11*y^2 = -2 |     3 |     1
  4 | x^2 - 19*y^2 = -2 |    13 |     3
  5 | x^2 - 23*y^2 = +2 |     5 |     1
  6 | x^2 - 31*y^2 = +2 |    39 |     7
  7 | x^2 - 43*y^2 = -2 |    59 |     9
  8 | x^2 - 47*y^2 = +2 |     7 |     1
  9 | x^2 - 59*y^2 = -2 |    23 |     3

Cf. A002145, A306529 (x-values).

Similar sequences: A094048, A094049 (x^2 - A002144(n)*y^2 = -1); A306618, A306619 (2*x^2 - A002145(n)*y^2 = (-1)^((p+1)/4)).

b(p) = if(isprime(p)&&p%4==3, y=1; while(!issquare(p*y^2 + 2*(-1)^((p+1)/4)), y++); y)
forprime(p=3, 500, if(p%4==3, print1(b(p), ", ")))

If the continued fraction of sqrt(A002145(n)) is [a_0; {a_1, a_2, ..., a_(k-1), a_k, a_(k-1), ..., a_1, 2*a_0}], where {} is the periodic part, let x/y = [a_0; a_1, a_2, ..., a_(k-1)], gcd(x, y) = 1, then A306529(n) = x and a(n) = y.

A306618 x-value of the smallest solution to 2x^2 - py^2 = (-1)^((p+1)/4), p = A002145(n).

Original entry on oeis.org

1, 2, 7, 3, 78, 4, 51, 732, 277, 191, 6, 44, 20621, 122, 416941, 8, 5123, 25, 1034, 9, 3993882, 210107, 203100, 10, 1325, 5248, 65030839, 20107956, 30953, 4584105462, 1036, 4889, 295081, 58746, 20725, 98465863939, 1494439626, 1612, 10173, 6040149252, 102607, 9460742124

Offset: 1

Jianing Song, Mar 25 2019

nonn

a(n) exists for all n.

X = 4*a(n)^2 - (-1)^((p+1)/4), Y = 2*a(n)*A306619(n) gives the smallest solution to x^2 - 2p*y^2 = 1, p = A002145(n).

			The smallest solution to 2*x^2 - p*y^2 = (-1)^((p+1)/4) for the first primes congruent to 3 modulo 4:
  n |       Equation      | x_min | y_min
  1 | 2*x^2 -  3*y^2 = -1 |     1 |     1
  2 | 2*x^2 -  7*y^2 = +1 |     2 |     1
  3 | 2*x^2 - 11*y^2 = -1 |     7 |     3
  4 | 2*x^2 - 19*y^2 = -1 |     3 |     1
  5 | 2*x^2 - 23*y^2 = +1 |    78 |    23
  6 | 2*x^2 - 31*y^2 = +1 |     4 |     1
  7 | 2*x^2 - 43*y^2 = -1 |    51 |    11
  8 | 2*x^2 - 47*y^2 = +1 |   732 |   151
  9 | 2*x^2 - 59*y^2 = -1 |   277 |    51

Cf. A002145, A306619 (y-values).

Similar sequences: A094048, A094049 (x^2 - A002144(n)*y^2 = -1); A306529, A306566 (x^2 - A002145(n)*y^2 = 2*(-1)^((p+1)/4)).

b(p) = if(isprime(p)&&p%4==3, x=1; while(!issquare((2*x^2 - (-1)^((p+1)/4))/p), x++); x)
forprime(p=3, 250, if(p%4==3, print1(b(p), ", ")))

If the continued fraction of sqrt(2*A002145(n)) is [a_0; {a_1, a_2, ..., a_(k-1), a_k, a_(k-1), ..., a_1, 2*a_0}], where {} is the periodic part, let x/y = [a_0; a_1, a_2, ..., a_(k-1)], gcd(x, y) = 1, then a(n) = x/2 and A306619(n) = y.

A306619 y-value of the smallest solution to 2x^2 - py^2 = (-1)^((p+1)/4), p = A002145(n).

Original entry on oeis.org

1, 1, 3, 1, 23, 1, 11, 151, 51, 33, 1, 7, 3201, 17, 57003, 1, 633, 3, 119, 1, 437071, 22209, 20783, 1, 129, 497, 6104097, 1839433, 399752993, 89, 411, 23817, 4711, 1611, 7475426163, 111543983, 119, 739, 436478927, 7089, 644468311, 103, 93487270491, 573497, 57, 4182991

Offset: 1

Jianing Song, Mar 25 2019

nonn

a(n) exists for all n.

X = 4*A306618(n)^2 - (-1)^((p+1)/4), Y = 2*A306618(n)*a(n) gives the smallest solution to x^2 - 2p*y^2 = 1, p = A002145(n).

			The smallest solution to 2*x^2 - p*y^2 = (-1)^((p+1)/4) for the first primes congruent to 3 modulo 4:
  n |       Equation      | x_min | y_min
  1 | 2*x^2 -  3*y^2 = -1 |     1 |     1
  2 | 2*x^2 -  7*y^2 = +1 |     2 |     1
  3 | 2*x^2 - 11*y^2 = -1 |     7 |     3
  4 | 2*x^2 - 19*y^2 = -1 |     3 |     1
  5 | 2*x^2 - 23*y^2 = +1 |    78 |    23
  6 | 2*x^2 - 31*y^2 = +1 |     4 |     1
  7 | 2*x^2 - 43*y^2 = -1 |    51 |    11
  8 | 2*x^2 - 47*y^2 = +1 |   732 |   151
  9 | 2*x^2 - 59*y^2 = -1 |   277 |    51

Cf. A002145, A306618 (x-values).

Similar sequences: A094048, A094049 (x^2 - A002144(n)*y^2 = -1); A306529, A306566 (x^2 - A002145(n)*y^2 = 2*(-1)^((p+1)/4)).

b(p) = if(isprime(p)&&p%4==3, y=1; while(!issquare((p*y^2 + (-1)^((p+1)/4))/2), y++); y)
forprime(p=3, 250, if(p%4==3, print1(b(p), ", ")))

If the continued fraction of sqrt(2*A002145(n)) is [a_0; {a_1, a_2, ..., a_(k-1), a_k, a_(k-1), ..., a_1, 2*a_0}], where {} is the periodic part, let x/y = [a_0; a_1, a_2, ..., a_(k-1)], gcd(x, y) = 1, then A306618(n) = x/2 and a(n) = y.

cp's OEIS Frontend

A094049 Let p(n) be the n-th prime congruent to 1 mod 4. Then a(n) = the least k for which m^2+1=p(n)*k^2 has a solution.

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A191860 First member of a pair of numbers occurring in the definition of 1-happy couples.

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Haskell

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A130226 Smallest integer x satisfying the Pell equation x^2-k*y^2=-1 for the values of k given in A031396.

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Maple

Mathematica

A306529 x-value of the smallest solution to x^2 - p*y^2 = 2*(-1)^((p+1)/4), p = A002145(n).

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A306566 y-value of the smallest solution to x^2 - p*y^2 = 2*(-1)^((p+1)/4), p = A002145(n).

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A306618 x-value of the smallest solution to 2*x^2 - p*y^2 = (-1)^((p+1)/4), p = A002145(n).

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A306619 y-value of the smallest solution to 2*x^2 - p*y^2 = (-1)^((p+1)/4), p = A002145(n).

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A306529 x-value of the smallest solution to x^2 - py^2 = 2(-1)^((p+1)/4), p = A002145(n).

A306566 y-value of the smallest solution to x^2 - py^2 = 2(-1)^((p+1)/4), p = A002145(n).

A306618 x-value of the smallest solution to 2x^2 - py^2 = (-1)^((p+1)/4), p = A002145(n).

A306619 y-value of the smallest solution to 2x^2 - py^2 = (-1)^((p+1)/4), p = A002145(n).