A078357
Minimal positive solution x of Pell equation y^2 - A077426(n)*x^2 = -4.
Original entry on oeis.org
1, 1, 2, 1, 2, 10, 1, 5, 2, 250, 1, 106, 1138, 2, 25, 146, 1, 298, 2, 5, 17, 1, 97, 10, 253970, 2, 1, 3034, 9148450, 2, 746, 10, 157, 126890, 1, 14341370, 5, 2, 110671282, 986, 7586, 1, 530, 130, 173, 2, 11068353370, 21685, 26, 694966754, 1, 17883410, 5528222698, 17, 87922, 2, 5, 41
Offset: 1
- O. Perron, "Die Lehre von den Kettenbruechen, Bd.I", Teubner, 1954, 1957 (Sec. 30, Satz 3.35, p. 109 and table p. 108).
-
$MaxExtraPrecision = 100; A077426 = Select[Range[ 600], ! IntegerQ[Sqrt[#]] && OddQ[ Length[ ContinuedFraction[(Sqrt[#] + 1)/2] // Last]] &]; a[n_] := {y, x} /. {ToRules[ Reduce[y > 0 && x > 0 && y^2 - A077426[[n]]*x^2 == -4, {y, x}, Integers] /. C[1] -> 0]} // Sort // First // Last; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Jun 21 2013 *)
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
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).
-
$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 *)
A078370
a(n) = 4*(n+1)*n + 5.
Original entry on oeis.org
5, 13, 29, 53, 85, 125, 173, 229, 293, 365, 445, 533, 629, 733, 845, 965, 1093, 1229, 1373, 1525, 1685, 1853, 2029, 2213, 2405, 2605, 2813, 3029, 3253, 3485, 3725, 3973, 4229, 4493, 4765, 5045, 5333, 5629, 5933, 6245, 6565, 6893, 7229, 7573, 7925, 8285, 8653, 9029
Offset: 0
- L. B. W. Jolley, "Summation of Series", Dover Publications, 1961, p. 176.
Subsequence of
A077426 (D values (not a square) for which Pell x^2 - D*y^2 = -4 is solvable in positive integers).
-
[4*n^2+4*n+5 : n in [0..80]]; // Wesley Ivan Hurt, Aug 29 2022
-
Table[4 n (n + 1) + 5, {n, 0, 45}] (* or *)
Table[8 Binomial[n + 1, 2] + 5, {n, 0, 45}] (* or *)
CoefficientList[Series[(5 - 2 x + 5 x^2)/(1 - x)^3, {x, 0, 45}], x] (* Michael De Vlieger, Jan 04 2017 *)
-
a(n)=4*n^2+4*n+5 \\ Charles R Greathouse IV, Sep 24 2015
-
a= lambda n: 4*n**2+4*n+5 # Indranil Ghosh, Jan 04 2017
-
(1 to 99 by 2).map(n => n * n + 4) // Alonso del Arte, May 29 2019
A077428
Minimal (positive) solution a(n) of Pell equation a(n)^2 - D(n)*b(n)^2 = +4 with D(n)= A077425(n). The companion sequence is b(n)=A078355(n).
Original entry on oeis.org
3, 11, 66, 5, 27, 46, 146, 4098, 7, 51, 302, 1523, 258, 25, 4562498, 9, 83, 1000002, 29, 125619266, 402, 82, 68123, 2408706, 11, 123, 33710, 173, 12166146, 190, 578, 3723, 4354, 45371, 23550, 13, 171, 124846, 1703027, 18498, 110, 12448646853698, 786
Offset: 1
- O. Perron, "Die Lehre von den Kettenbruechen, Bd.I", Teubner, 1954, 1957 (Sec. 30, Satz 3.35, p. 109 and table p. 108).
-
d = Select[Range[5, 300, 4], !IntegerQ[Sqrt[#]]&]; a[n_] := Module[{a, b, r}, a /. {r = Reduce[a > 0 && b > 0 && a^2 - d[[n]]*b^2 == 4, {a, b}, Integers]; (r /. C[1] -> 0) || (r /. C[1] -> 1) // ToRules} // Select[#, IntegerQ, 1] &] // First; Table[a[n], {n, 1, 43}] (* Jean-François Alcover, Jul 30 2013 *)
A077425
a(n) == 1 (mod 4) (see A016813), but not a square (i.e., not in A000290).
Original entry on oeis.org
5, 13, 17, 21, 29, 33, 37, 41, 45, 53, 57, 61, 65, 69, 73, 77, 85, 89, 93, 97, 101, 105, 109, 113, 117, 125, 129, 133, 137, 141, 145, 149, 153, 157, 161, 165, 173, 177, 181, 185, 189, 193, 197, 201, 205, 209, 213, 217, 221, 229, 233, 237, 241, 245, 249, 253, 257
Offset: 1
-
A077425 := proc(n::integer) local resul,i ; resul := 5 ; i := 1 ; while i < n do resul := resul+4 ; while issqr(resul) do resul := resul+4 ; od ; i:= i+1 ; od ; RETURN(resul) ; end proc:
seq(A077425(n),n=1..31) ; # R. J. Mathar, Apr 25 2006
-
Select[Range[5,300,4],!IntegerQ[Sqrt[#]]&] (* Harvey P. Dale, Dec 05 2012 *)
-
[n | n <- vector(100,n,4*n+1), !issquare(n)] \\ Charles R Greathouse IV, Mar 11 2014
-
list(lim)=my(v=List()); for(s=2,sqrtint((lim\=1)+1), forstep(n=s^2 + if(s%2,4,1), min((s+1)^2-1,lim), 4, listput(v,n))); Vec(v) \\ Charles R Greathouse IV, Nov 04 2021
-
from operator import sub
from sympy import integer_nthroot
def A077425(n): return n+sub(*integer_nthroot(n,2))<<2|1 # Chai Wah Wu, Oct 01 2024
A077427
Primitive period length of (regular) continued fraction of (sqrt(D(n))+1)/2 for D(n)=A077425(n).
Original entry on oeis.org
1, 1, 3, 2, 1, 4, 3, 5, 2, 1, 6, 3, 3, 4, 9, 2, 1, 7, 2, 9, 3, 6, 7, 7, 2, 1, 10, 4, 7, 4, 3, 5, 8, 5, 10, 2, 1, 12, 5, 3, 4, 15, 3, 14, 4, 12, 4, 16, 2, 1, 9, 2, 19, 2, 16, 6, 3, 8, 11, 5, 6, 9, 15, 2, 1, 10, 10, 4, 6, 19, 3, 4, 3, 16
Offset: 1
a(6)=4 because the (periodic) continued fraction for (sqrt(D(6))+1)/2 = (sqrt(33)+1)/2 = 3.372281324... is [3, periodic(2, 1, 2, 5,)] with period length 4. Because these continued fractions are always of the form [b(0),periodic(b(1),b(2),...,b(2),b(1),2*b(0)-1,)] with the symmetric piece b(1),b(2),..., b(2),b(1), Perron op. cit. writes for this b(0),b(1),b(2),...,(b(k/2)) if the period length k is even and b(0),b(1),b(2),...,b((k-1)/2) if the period length is odd. In this example: k=4 and Perron writes 3,2,(1). Another example: D(8)= A077425(8)=41 leads to Perron's 3,1,2 standing for [3,periodic(1,2,2,1,5,)], the continued fraction for (sqrt(41)+1)/2 which has odd period length a(8)=5.
a(4)=2 is even and D(4)=A077425(4)=21, hence x^2 - 21*y^2 = -4 has no nontrivial integer solution.
a(8)=5 is odd and D(8)=A077425(8)=41, hence x^2 - 41*y^2 = -4 is solvable (with nontrivial integers) as well as x^2 - 41*y^2 = +4.
- O. Perron, "Die Lehre von den Kettenbruechen, Bd.I", Teubner, 1954, 1957 (Sec. 30, Satz 3.35, p. 109).
A078361
Minimal positive solution a(n) of Pell equation a(n)^2 - D(n)*b(n)^2 = +4 or -4 with D(n)=A077425(n). The companion sequence is b(n)=A077058(n).
Original entry on oeis.org
1, 3, 8, 5, 5, 46, 12, 64, 7, 7, 302, 39, 16, 25, 2136, 9, 9, 1000, 29, 11208, 20, 82, 261, 1552, 11, 11, 33710, 173, 3488, 190, 24, 61, 4354, 213, 23550, 13, 13, 124846, 1305, 136, 110, 3528264, 28, 1030190, 43, 93102, 73, 7688126, 15, 15, 46312, 77
Offset: 1
29=D(5)=A077425(5) is A077426(4), hence a(5)=5 and b(5)=A077058(5)=1 solve a^2 - 29*b^2=-4 minimally and a(+,5)=a(5)^2+2=27 with b(+,5)=a(5)*b(5)=5*1=5 solve a^2 - 29*b^2=+4 minimally. See also A077428 with companion A078355.
21=D(4)=A077425(4) is not in A077426, hence a(4)=5 and b(4)=A077058(4)=1 give the solution with minimal positive b of a^2 - 21*b^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).
A175155
Numbers m satisfying m^2 + 1 = x^2 * y^3 for positive integers x and y.
Original entry on oeis.org
0, 682, 1268860318, 1459639851109444, 2360712083917682, 86149711981264908618, 4392100110703410665318, 8171493471761113423918890682, 15203047261220215902863544865414318, 5484296027914919579181500526692857773246, 28285239023397517753374058381589688919682, 12439333951782387734360136352377558500557329868
Offset: 1
For m=682, m^2 + 1 = 465125 = 61^2 * 5^3.
- Albert H. Beiler, "The Pellian" (Chap. 22), Recreations in the Theory of Numbers, 2nd ed. NY: Dover, 1966.
- 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.
- J. M. De Koninck, Ces nombres qui nous fascinent, Ellipses, 2008, p. 108.
-
C:=array(0..20,0..20):C[1,1]=1: C[2,1]=1: n1:=682:x1:=61:for nn from 1 by 2 to 15 do:s:=0:for i from 2 to 15 do:for j from 1 to i do:C[i,j]:= C[i-1,j] + C[i-1,j-1]: od:od:for n from 1 by 2 to nn+1 do:s:=s + C[nn+1,n] * n1^(nn-n+1)*x1^(n-1)*125^((n-1)/2):od:print (s):od: # Michel Lagneau
# 2nd program R. J. Mathar, Mar 16 2016:
# print (nonsorted!) all solutions of A175155 up to search limit
with(numtheory):
# upper limit for solutions n
nsearchlim := 10^40 :
A175155y := proc(y::integer)
local disc;
disc := y^3 ;
cfrac(sqrt(disc),periodic,quotients) ;
end proc:
for y from 2 do
if issqrfree(y) then
# find continued fraction for x^2-(y^3=disc)*y^2=-1, sqrt(disc)
cf := A175155y(y) ;
nlen := nops(op(2,cf)) ;
if type(nlen,odd) then
# fundamental solution
fuso := numtheory[nthconver](cf,nlen-1) ;
fusolx := numer(fuso) ;
fusoly := denom(fuso) ;
solx := fusolx ;
soly := fusoly ;
while solx <= nsearchlim do
rhhs := solx^2-y^3*soly^2 ;
if rhhs = -1 then
# print("n=",solx,"x=",soly,"y=",y^3) ;
print(solx) ;
end if;
# solutions from fundamental solution
tempx := fusolx*solx+y^3*fusoly*soly ;
tempy := fusolx*soly+fusoly*solx ;
solx := tempx ;
soly := tempy ;
end do;
end if;
fi;
end do:
-
nmax = 10^50; ymax = 100; instances = 10; fi[y_] := n /. FindInstance[0 <= n <= nmax && x > 0 && n^2 + 1 == x^2*y^3, {n, x}, Integers, instances]; yy = Select[Range[1, ymax, 2], !IntegerQ[Sqrt[#]] && OddQ[ Length[ ContinuedFraction[Sqrt[#]][[2]]]]&]; Join[{0}, fi /@ yy // Flatten // Union // Most] (* Jean-François Alcover, Jul 12 2017 *)
-
is(n)=ispowerful(n^2+1) \\ Charles R Greathouse IV, Nov 16 2012
Added condition that x and y must be positive. Added missing initial term 0. Added warning that b-file has not been proved to be correct - there could be missing entries. -
N. J. A. Sloane, Nov 17 2012
A225432
Twice the coefficient of sqrt(q) in e^h, where e is the fundamental unit and h is the class number of Q(sqrt(q)), q prime and congruent to 1 mod 4. (The coefficient lies in (1/2)Z, so twice it is an integer.)
Original entry on oeis.org
1, 1, 2, 1, 2, 10, 1, 5, 250, 106, 1138, 2, 25, 146, 298, 5, 17, 1, 97, 253970, 2, 226, 3034, 9148450, 2050, 10, 157, 126890, 1, 14341370, 5, 110671282, 986, 7586, 530, 130, 173, 5129602, 11068353370, 21685, 694966754, 17883410, 5528222698, 17, 41, 11248618, 60037, 10, 242718010, 24514292738
Offset: 1
- R. R. Bruner and J. P. C. Greenlees, The Connective K-theory of Finite Groups, Memoirs AMS, Vol. 165, No. 785, 2003.
- T. Mitsuhiro, T. Nakahara and T. Uehara, The Class Number Formula of a Real Quadratic Field and an Estimate of the Value of a Unit, Canadian Mathematical Bulletin, 38(1)(1995), 98-103.
-
// Magma code to generate all terms for which the prime q is less than or equal to 4N+1 (an initial segment of the sequence). (Note that the brute force computation of the fundamental unit is very inefficient, and will have trouble producing the 39th term.)
N := 40;
pr := [4*n+1 : n in [1..N] | IsPrime(4*n+1)];
for i in [1..#pr] do
q := pr[i];
Q := QuadraticField(q);
h := ClassNumber(Q);
x := 1;
while not IsSquare(x*x*q-4) do
x := x+1;
end while;
x := x/2;
tr,y := IsSquare(x*x*q-1);
e := y + x*s;
eh := e^h;
b := (eh-Trace(eh)/2)/s;
print i,2*b;
end for;
-
(* e.g., first 270 terms *)
Lq = Select[4*Range[1000] + 1, PrimeQ[#] &];
Lh = NumberFieldClassNumber[Sqrt[Lq]];
Le = NumberFieldFundamentalUnits[Sqrt[Lq]];
Transpose[RootReduce[(Le^(2 Lh) + 1)/(Sqrt[Lq] Le^Lh)]][[1]]
(* Zichang Wang, Dec 15 2022 *)
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