A200231 -id:A200231 - cp's OEIS frontend

A199949 Decimal expansion of least x satisfying x^2 + cos(x) = 2*sin(x).

6, 5, 9, 2, 6, 6, 0, 4, 5, 7, 6, 6, 9, 4, 6, 0, 7, 4, 5, 3, 7, 3, 4, 8, 5, 7, 9, 5, 6, 3, 0, 6, 7, 6, 1, 1, 6, 1, 5, 3, 2, 8, 0, 2, 1, 6, 4, 4, 5, 1, 6, 7, 9, 7, 3, 6, 0, 9, 4, 5, 1, 3, 0, 3, 1, 4, 1, 0, 7, 3, 6, 4, 4, 5, 5, 8, 7, 4, 2, 6, 6, 2, 4, 4, 0, 7, 1, 9, 5, 1, 9, 3, 1, 6, 4, 1, 4, 4, 7

Offset: 0

Clark Kimberling, Nov 12 2011

For many choices of a,b,c, there are exactly two numbers x>0 satisfying a*x^2+b*cos(x)=c*sin(x).
Guide to related sequences, with graphs included in Mathematica programs:
a.... b.... c.... least x, greatest x
... 1.... 2.... A199949, A199950
... 1.... 3.... A199951, A199952
... 1.... 4.... A199953, A199954
... 2.... 3.... A199955, A199956
... 2.... 4.... A199957, A199958
... 3.... 3.... A199959, A199960
... 3.... 4.... A199961, A199962
... 4.... 3.... A199963, A199964
... 4.... 4.... A199965, A199966
... 1.... 3.... A199967, A200003
... 1.... 4.... A200004, A200005
... 1.... 4.... A200006, A200007
... 1.... 4.... A200008, A200009
.. -1.... 1.... A200010, A200011
.. -1.... 2.... A200012, A200013
.. -1.... 3.... A200014, A200015
.. -1.... 4.... A200016, A200017
.. -2.... 1.... A200018, A200019
.. -2.... 2.... A200020, A200021
.. -2.... 3.... A200022, A200023
.. -2.... 4.... A200024, A200025
.. -3.... 1.... A200026, A200027
.. -3.... 2.... A200093, A200094
.. -3.... 3.... A200095, A200096
.. -3.... 4.... A200097, A200098
.. -4.... 1.... A200099, A200100
.. -4.... 2.... A200101, A200102
.. -4.... 3.... A200103, A200104
.. -4.... 4.... A200105, A200106
.. -1.... 1.... A200107, A200108
.. -1.... 2.... A200109, A200110
.. -1.... 3.... A200111, A200112
.. -1.... 4.... A200114, A200115
.. -2.... 1.... A200116, A200117
.. -2.... 3.... A200118, A200119
.. -3.... 1.... A200120, A200121
.. -3.... 2.... A200122, A200123
.. -3.... 3.... A200124, A200125
.. -3.... 4.... A200126, A200127
.. -4.... 1.... A200128, A200129
.. -4.... 3.... A200130, A200131
.. -1.... 1.... A200132, A200133
.. -1.... 2.... A200223, A200224
.. -1.... 3.... A200225, A200226
.. -1.... 4.... A200227, A200228
.. -2.... 1.... A200229, A200230
.. -2.... 2.... A200231, A200232
.. -2.... 3.... A200233, A200234
.. -2.... 4.... A200235, A200236
.. -3.... 1.... A200237, A200238
.. -3.... 2.... A200239, A200240
.. -3.... 4.... A200241, A200242
.. -4.... 1.... A200277, A200278
.. -4.... 2.... A200279, A200280
.. -4.... 3.... A200281, A200282
.. -4.... 4.... A200283, A200284
.. -1.... 1.... A200285, A200286
.. -1.... 2.... A200287, A200288
.. -1.... 3.... A200289, A200290
.. -1.... 4.... A200291, A200292
.. -2.... 1.... A200293, A200294
.. -2.... 3.... A200295, A200296
.. -3.... 1.... A200299, A200300
.. -3.... 2.... A200297, A200298
.. -3.... 3.... A200301, A200302
.. -3.... 4.... A200303, A200304
.. -4.... 1.... A200305, A200306
.. -4.... 3.... A200307, A200308
Suppose that f(x,u,v) is a function of three real variables and that g(u,v) is a function defined implicitly by f(g(u,v),u,v)=0.  We call the graph of z=g(u,v) an implicit surface of f.
For an example related to A199949, take f(x,u,v)=x^2+u*cos(x)-v*sin(x) and g(u,v) = a nonzero solution x of f(x,u,v)=0.  If there is more than one nonzero solution, care must be taken to ensure that the resulting function g(u,v) is single-valued and continuous.  A portion of an implicit surface is plotted by Program 2 in the Mathematica section.

			least x:  0.659266045766946074537348579563067611...
greatest x: 1.2710268008159460640047188480978502...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Index entries for transcendental numbers.

Cf. A199950.

(* Program 1:  A199949 *)
a = 1; b = 1; c = 2;
f[x_] := a*x^2 + b*Cos[x]; g[x_] := c*Sin[x]
Plot[{f[x], g[x]}, {x, -1, 2}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, .65, .66}, WorkingPrecision -> 110]
RealDigits[r]   (* A199949 *)
r = x /. FindRoot[f[x] == g[x], {x, 1.27, 1.28}, WorkingPrecision -> 110]
RealDigits[r]   (* A199950 *)
(* Program 2: implicit surface of x^2+u*cos(x)=v*sin(x) *)
f[{x_, u_, v_}] := x^2 + u*Cos[x] - v*Sin[x];
t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 0, 1}]}, {u, -5, 0}, {v, 0, 1}];
ListPlot3D[Flatten[t, 1]]  (* for A199949 *)

a=1; b=1; c=2; solve(x=0, 1, a*x^2 + b*cos(x) - c*sin(x)) \\ G. C. Greubel, Jul 05 2018

A-number corrected by Jaroslav Krizek, Nov 27 2011

A200232 Decimal expansion of greatest x satisfying 3x^2-2cos(x)=2*sin(x).

Original entry on oeis.org

9, 6, 3, 2, 9, 1, 3, 7, 6, 6, 1, 9, 6, 7, 9, 1, 0, 4, 6, 5, 5, 6, 4, 1, 8, 2, 9, 6, 6, 4, 1, 6, 4, 2, 3, 0, 2, 3, 0, 3, 9, 1, 9, 8, 6, 7, 2, 7, 0, 6, 2, 5, 1, 0, 1, 2, 1, 4, 5, 6, 4, 6, 1, 4, 8, 6, 2, 1, 1, 4, 2, 3, 8, 3, 8, 8, 9, 9, 3, 5, 1, 0, 0, 4, 1, 0, 9, 7, 1, 9, 4, 2, 1, 6, 6, 1, 4, 4, 8

Offset: 0

Clark Kimberling, Nov 14 2011

See A199949 for a guide to related sequences. The Mathematica program includes a graph.

			least x: -0.508066683701868134653059484203509821...
greatest x: 0.9632913766196791046556418296641642...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Index entries for transcendental numbers.

Cf. A199949.

a = 3; b = -2; c = 2;
f[x_] := a*x^2 + b*Cos[x]; g[x_] := c*Sin[x]
Plot[{f[x], g[x]}, {x, -2, 2}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, -.51, -.50}, WorkingPrecision -> 110]
RealDigits[r]   (* A200231 *)
r = x /. FindRoot[f[x] == g[x], {x, .96, .97}, WorkingPrecision -> 110]
RealDigits[r]   (* A200232 *)

a=3; b=-2; c=2; solve(x=.96, .97, a*x^2 + b*cos(x) - c*sin(x)) \\ G. C. Greubel, Jun 22 2018

A247607 Primes whose number of symmetric connections to other primes within the same power of two interval exceeds the number of symmetric connections to composites.

Original entry on oeis.org

5, 7, 17, 19, 29, 31, 43, 59, 103, 281, 457, 461, 463, 499, 607, 1409, 1451, 2143, 2657, 4229, 16063, 19583, 19699, 62143, 65537, 70919, 107347, 113159, 124783, 124981, 600703, 3103423, 18936719

Offset: 1

Brad Clardy, Sep 22 2014

nonn

While there may be some additional terms, it is thought to be a finite sequence. An exhaustive search was conducted up to 2^29.

Members larger than 2^8 will also be in A200321.

			In the interval (2^4,2^5) [17,19,21,23,25,27,29,31], the prime 17 symmetrically couples with 31 around the midpoint of the interval, 23 around the midpoint of the halved interval, and 19 in the quartered interval. There is no composite couple. It has 3 prime and 0 composite symmetric connections. The same process for 19 produces 2 prime and 1 composite connection. Therefore 17 and 19 are members. 23 couples with 25, 17 and 22. It has 1 prime and 2 composite connections so it is not a member.

Cf. A200231.

XOR := func;
function PCcoord(X,i,P,C)
if (i eq 1) then
    if (P gt C) then return true;
      else
      return false,P,C;
    end if;
  else
    xornum:=2^i - 2;
    xorcouple:=XOR(X, xornum);
    if (IsPrime(xorcouple)) then
       return PCcoord(X, i-1,P+1,C);
    else
       return PCcoord(X, i-1,P,C+1);
    end if;
  end if;
end function;
for k:= 1 to 2^10 + 1 by 2 do
  if IsPrime(k) then
     if PCcoord(k,Ilog2(k),0,0) then k;
     end if;
  end if;
end for;

cp's OEIS Frontend

A199949 Decimal expansion of least x satisfying x^2 + cos(x) = 2*sin(x).

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A200232 Decimal expansion of greatest x satisfying 3x^2-2cos(x)=2*sin(x).

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A247607 Primes whose number of symmetric connections to other primes within the same power of two interval exceeds the number of symmetric connections to composites.

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Magma

cp's OEIS Frontend

A199949 Decimal expansion of least x satisfying x^2 + cos(x) = 2*sin(x).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A200232 Decimal expansion of greatest x satisfying 3*x^2-2*cos(x)=2*sin(x).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A247607 Primes whose number of symmetric connections to other primes within the same power of two interval exceeds the number of symmetric connections to composites.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Magma

A200232 Decimal expansion of greatest x satisfying 3x^2-2cos(x)=2*sin(x).