A077427 -id:A077427 - cp's OEIS frontend

A078358 Non-oblong numbers: Complement of A002378.

1, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74

Offset: 1

Wolfdieter Lang, Nov 29 2002

The (primitive) period length k(n)=A077427(n) of the (regular) continued fraction of (sqrt(4*a(n)+1)+1)/2 determines whether or not the Diophantine equation (2*x-y)^2 - (1+4*a(n))*y^2 = +4 or -4 is solvable and the approximants of this continued fraction give all solutions. See A077057.
The following sequences all have the same parity: A004737, A006590, A027052, A071028, A071797, A078358, A078446. - Jeremy Gardiner, Mar 16 2003
Infinite series 1/A078358(n) is divergent. Proof: Harmonic series 1/A000027(n) is divergent and can be distributed on two subseries 1/A002378(k+1) and 1/A078358(m). The infinite subseries 1/A002378(k+1) is convergent to 1, so Sum_{n>=1} 1/A078358(n) is divergent. - Artur Jasinski, Sep 28 2008

O. Perron, "Die Lehre von den Kettenbruechen, Bd.I", Teubner, 1954, 1957 (Sec. 30, Satz 3.35, p. 109 and table p. 108).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Oskar Perron, Die Lehre von den Kettenbrüchen, Teubner, Leipzig, 1913.

a(n)=(A077425(n)-1)/4.

Cf. A049068 (subsequence), A144786.

a078358 n = a078358_list !! (n-1)
a078358_list = filter ((== 0) . a005369) [0..]
-- Reinhard Zumkeller, Jul 04 2014, May 08 2012

Complement[Range[930], Table[n (n + 1), {n, 0, 30}]] (* and *) Table[Ceiling[Sqrt[n]] + n - 1, {n, 900}] (* Vladimir Joseph Stephan Orlovsky, Jul 20 2011 *)

a(n)=sqrtint(n-1)+n \\ Charles R Greathouse IV, Jan 17 2013

from operator import sub
from sympy import integer_nthroot
def A078358(n): return n+sub(*integer_nthroot(n,2)) # Chai Wah Wu, Oct 01 2024

4*a(n)+1 is not a square number.
a(n) = ceiling(sqrt(n)) + n -1. - Leroy Quet, Jul 06 2007
A005369(a(n)) = 0. - Reinhard Zumkeller, Jul 05 2014

A077426 Nonsquare integers n such that the continued fraction (sqrt(n)+1)/2 has odd period length.

Original entry on oeis.org

5, 13, 17, 29, 37, 41, 53, 61, 65, 73, 85, 89, 97, 101, 109, 113, 125, 137, 145, 149, 157, 173, 181, 185, 193, 197, 229, 233, 241, 257, 265, 269, 277, 281, 293, 313, 317, 325, 337, 349, 353, 365, 373, 389, 397, 401, 409, 421, 425, 433, 445, 449, 457, 461, 481, 485

Offset: 1

Wolfdieter Lang, Nov 29 2002

Nonsquare integers n for which Pell equation x^2 - n*y^2 = -4 has infinitely many integer solutions. The smallest solutions are given in A078356 and A078357.

O. Perron, "Die Lehre von den Kettenbruechen, Bd.I", Teubner, 1954, 1957 (Sec. 30, table p. 108).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Wikipedia, Pell's equation

A subsequence of A077425.
Odd elements of A003814.
Cf. A077427, A172000.

isOddPrim := proc(n::integer)
    local cf;
    cf := numtheory[cfrac]((sqrt(n)+1)/2,'periodic','quotients') ;
    if nops(op(2,cf)) mod 2 =1 then
        RETURN(true) ;
    else
        RETURN(false) ;
    fi ;
end:
notA077426 := proc(n::integer)
    if issqr(n) then
        RETURN(true) ;
    elif not isOddPrim(n) then
        RETURN(true) ;
    else
        RETURN(false) ;
    fi ;
end:
A077426 := proc(n::integer)
    local resul,i ;
    resul := 5 ;
    i := 1 ;
    while i < n do
        resul := resul+4 ;
        while notA077426(resul) do
            resul := resul+4 ;
        od ;
        i:= i+1 ;
    od ;
    RETURN(resul) ;
end:
for n from 1 to 61 do print(A077426(n)) ; od : # R. J. Mathar, Apr 25 2006

fQ[n_] := !IntegerQ@ Sqrt@ n && OddQ@ Length@ ContinuedFraction[(Sqrt@ n + 1)/2][[2]]; Select[Range@ 500, fQ] (* Robert G. Wilson v, Nov 17 2012 *)

Edited and extended by Max Alekseyev, Mar 03 2010

Edited by Max Alekseyev, Mar 05 2010

A078356 Minimal positive solution z of Pell equation z^2 - A077426(n)*t^2 = -4.

Original entry on oeis.org

1, 3, 8, 5, 12, 64, 7, 39, 16, 2136, 9, 1000, 11208, 20, 261, 1552, 11, 3488, 24, 61, 213, 13, 1305, 136, 3528264, 28, 15, 46312, 142022136, 32, 12144, 164, 2613, 2127064, 17, 253724736, 89, 36, 2031654672, 18420, 142528, 19, 10236, 2564, 3447, 40, 223843593936

Offset: 1

Wolfdieter Lang, Nov 29 2002

nonn

The corresponding values of t are given in A078357.
Computed from Perron's table (see reference p. 108) which gives the minimal x,y values for the Diophantine equation x^2 - x*y - ((D(m)-1)/4)*y^2 = +1 and -1 for respectively D(m)=A077425(m) and D(m)=A077426(m) (this second case excludes in Perron's table the D values with a 'Teilnenner' in brackets).
The conversion from the x,y values of Perron's table to the minimal a=a(n) and b=b(n) solutions of a^2 - D(n)*b^2 =-4 see a comment in A077428. Here only D values with no 'Teilnenner' in brackets are of interest and a(n)=2*x(n)-y(n) and b(n)=y(n). E.g. D=41, with 'Teilnenner von (sqrt(D)+1)/2' in the notation, explained in an example of A077427, 3,1,2 (period length k=5) and (x,y)=(37,10) which translates to the minimal solution (a,b)=(64,10).
Generic D(n) values are those from A078370(k)=(4*k(k+1)+5), k>=0, which are 5 (mod 8). For such D values the minimal solution is (a,b)=(2*k+1,1) (e.g. D(7)= A077426(7) = 53 = A078370(3) with a(7)= 2*3+1=7 and b(7)=A078357(7)=1).
The general solution of Pell a^2-D(n)*b^2 = -4 with generic D(n)=A078370(k), k>=0, is a(n,m)= (2*k+1)*S(2*m,sqrt(D(n))) and b(n,m)= T(2*m+1,sqrt(D(n))/2)/(sqrt(D(n))/2), m>=0, with T(n,x), resp. S(n,x), Chebyshev's polynomials of the first, resp. second, kind. See A053120 resp. A049310.
For non-generic D(n) (not from A078370) the general solution of a^2-D(n)*b^2 = -4 is a(n,m)=a(n)*S(2*m,sqrt(a(n)^2+4)) and b(n,m)= b(n)*T(2*m+1,sqrt(a(n)^2+4)/2)/(sqrt(a(n)^2+4)/2), m>=0, with Chebyshev's polynomials and in this case b(n)>1.

			41=D(6)=A077426(6) (also A077425(8)), hence a(6)=64 and b(6)=A078357(6)=10 satisfies 64^2 - 41*10^2 = -4.

O. Perron, "Die Lehre von den Kettenbruechen, Bd.I", Teubner, 1954, 1957 (Sec. 30, Satz 3.35, p. 109 and table p. 108).

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Index entries for sequences related to Chebyshev polynomials.

$MaxExtraPrecision = 100; A077426 = Select[Range[ 500], ! IntegerQ[Sqrt[#]] && OddQ[ Length[ ContinuedFraction[(Sqrt[#] + 1)/2] // Last]] &]; a[n_] := {z, t} /. {ToRules[ Reduce[z > 0 && t > 0 && z^2 - A077426[[n]]*t^2 == -4, {z, t}, Integers] /. C[1] -> 0]} // Sort // First // First; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Jun 21 2013 *)

More terms from R. J. Mathar, Sep 24 2009

Edited by Max Alekseyev, Mar 03 2010

A077057 Minimal positive solution a(n) of Diophantine equation a(n)^2 - a(n)b(n) - G(n)b(n)^2 = +1 or -1 with G(n) := A078358(n). The companion sequence is b(n)=A077058(n).

Original entry on oeis.org

1, 2, 5, 3, 3, 27, 7, 37, 4, 4, 171, 22, 9, 14, 1193, 5, 5, 553, 16, 6173, 11, 45, 143, 849, 6, 6, 18339, 94, 1893, 103, 13, 33, 2353, 115, 12703, 7, 7, 67115, 701, 73, 59, 1891117, 15, 551427, 23, 49771, 39, 4105015, 8, 8, 24673, 41, 75585293, 25, 9095891, 989, 17, 386, 6445, 87, 771, 1385

Offset: 1

Wolfdieter Lang, Nov 29 2002

nonn

This equation can also be written as (2*a(n) - b(n))^2 - D(n)*b(n)^2 = +4 or -4 with D(n) := A077425(n) = 1 + 4*G(n).
This is from Perron's table (see reference p. 108, for n = 1..28) which gives the minimal x,y values which solve the above mentioned Diophantine equations.
For Pell equation x^2 - D*y^2 = +4, see A077428 and A078355. For Pell equation x^2 - D*y^2 = -4, see A078356 and A078357.

O. Perron, "Die Lehre von den Kettenbruechen, Bd.I", Teubner, 1954, 1957 (Sec. 30, Satz 3.35, p. 109 and table p. 108).

Wolfdieter Lang, Pairs of smallest positive solutions for +1 and -1 if they exist.

Cf. A077427, A217470.

g[n_] := Ceiling[Sqrt[n]] + n - 1; r[n_] := Reduce[an > 0 && bn > 0 && (an ^2 - an*bn - g[n]*bn^2 == 1 || an^2 - an*bn - g[n]*bn^2 == - 1), {an, bn}, Integers] /. C -> c; ab[n_] := DeleteCases[ Flatten[ Table[{an, bn} /. {ToRules[r[n]]} // Simplify, {c[1], 0, 1}], 1], an | bn]; a[n_] := a[n] = Min[ab[n][[All, 1]]]; Table[Print[{n, a[n]}]; a[n], {n, 1, 62}] (* Jean-François Alcover, Oct 04 2012 *)

a(n) = (A078361(n) + A077058(n)) / 2. [Max Alekseyev, Feb 06 2010]

More terms from Max Alekseyev, Feb 06 2010

a(9), a(33), a(54) corrected (after notice by Jean-François Alcover); a(58) through a(62) added. - Wolfdieter Lang, Oct 04 2012

A217470 The Diophantine equation x^2 - xy - Gy^2 = -1, G a positive integer, D = 4*G + 1 not a perfect square, has no solution precisely for G = a(n).

Original entry on oeis.org

5, 8, 11, 14, 17, 19, 23, 26, 29, 32, 33, 35, 38, 40, 41, 44, 47, 50, 51, 52, 53, 54, 55, 59, 61, 62, 63, 65, 68, 71, 74, 75, 76, 77, 80, 82, 83, 85, 86, 89, 92, 94, 95, 96, 98, 101

Offset: 1

Wolfdieter Lang, Oct 04 2012

nonn

See the Perron reference for the theorem which by negation implies that this quadratic Diophantine equation has no solution if and only if A077427 is even.
See the pairs (x, y) = (A077057, A077058) which for these a(n) values are the smallest positive solutions of the Diophantine equation x^2 - x*y - a(n)*y^ = +1.
In the table on p. 108 of the Perron reference these a(n) values, called there also G, are the ones were in the third column numbers in brackets appear.
The case D = 4*G + 1 = m^2 > 1 has trivially no solutions: the equation is then X^2 - Y^2 = -4, with  X = |2*x-y|, Y = |m*y|. X and Y are either both even or both odd. In the first case one is led to the equation v^2 - w^2 = (v-w)*(v+w)= -1, with X = 2*v and Y = 2*w, and there is only the solution (v,w) = (0,1), hence 2*x = y, m*y = 2. But then m=2 and y=1 with non-integer x solution. In the other case X = 2*v+1 and Y = 2*w+1, v not w, leading to v + w + 1 = -1 with no positive integer solution. Thanks to T. D. Noe for pointing out that one has to mention that these values G = A002378(k), k>=1, with D a perfect (odd) square, are here not included.

			a(1) = 5 because 5 = A078358(4) and A077427(4) = 2, which is even.

O. Perron, "Die Lehre von den Kettenbruechen, Bd.I", Teubner, 1954, 1957 (Sec. 30, Satz 3.35, p. 109 and table p. 108).

Cf. A077057, A077058, A078358, A077427.

a(n) gives the increasingly ordered values for G from A078358 which appear at position k where A077427(k) is even, for k>=1. The next even number in A077427 appears for k = 6 and

A078358(6) = 8, hence a(2) = 8.

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A078358 Non-oblong numbers: Complement of A002378.

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A077426 Nonsquare integers n such that the continued fraction (sqrt(n)+1)/2 has odd period length.

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A078356 Minimal positive solution z of Pell equation z^2 - A077426(n)*t^2 = -4.

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A077057 Minimal positive solution a(n) of Diophantine equation a(n)^2 - a(n)*b(n) - G(n)*b(n)^2 = +1 or -1 with G(n) := A078358(n). The companion sequence is b(n)=A077058(n).

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A217470 The Diophantine equation x^2 - x*y - G*y^2 = -1, G a positive integer, D = 4*G + 1 not a perfect square, has no solution precisely for G = a(n).

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A077057 Minimal positive solution a(n) of Diophantine equation a(n)^2 - a(n)b(n) - G(n)b(n)^2 = +1 or -1 with G(n) := A078358(n). The companion sequence is b(n)=A077058(n).

A217470 The Diophantine equation x^2 - xy - Gy^2 = -1, G a positive integer, D = 4*G + 1 not a perfect square, has no solution precisely for G = a(n).