cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 11-16 of 16 results.

A327136 Numbers k such that sin(2k) > sin(2k+2) < sin(2k+4).

Original entry on oeis.org

1, 4, 8, 11, 14, 17, 20, 23, 26, 30, 33, 36, 39, 42, 45, 48, 52, 55, 58, 61, 64, 67, 70, 74, 77, 80, 83, 86, 89, 92, 96, 99, 102, 105, 108, 111, 114, 118, 121, 124, 127, 130, 133, 136, 140, 143, 146, 149, 152, 155, 158, 162, 165, 168, 171, 174, 177, 180, 184
Offset: 1

Views

Author

Clark Kimberling, Aug 23 2019

Keywords

Comments

The sequences A026317, A327136, A327137 partition the nonnegative integers.
Conjecture: 1.285 < n*Pi - a(n) < 1.286 for n >= 1.

Examples

			(sin 2, sin 4, ...) = (0.9, -0.7, -0.2, 0.9, -0.5, ...) approximately, so that the differences, in sign, are - + + -  + + - - + - - + ..., with "+" in places 2,3,5,6,... (A026317), "- +" starting in places 1,4,8,11,... (A327136), and "- - +" starting in places 7,10,13,16,... (A327137).

Links

Crossrefs

Cf. A026309, A246303, A026317.

Programs

  • Mathematica
    z = 500; f[x_] := f[x] = Sin[2 x]; t = Range[1, z];
Select[t, f[#] < f[# + 1] &]    (* A026317 *)
Select[t, f[#] > f[# + 1] < f[# + 2] &]  (* A327136 *)
Select[t, f[#] > f[# + 1] > f[# + 2] < f[# + 3] &]   (* A327137 *)

A327137 Numbers k such that sin(2k) > sin(2k+2) > sin(2k+4) < sin(2k+6).

Original entry on oeis.org

7, 10, 13, 16, 29, 32, 35, 38, 51, 54, 57, 60, 73, 76, 79, 82, 95, 98, 101, 104, 117, 120, 123, 126, 139, 142, 145, 148, 161, 164, 167, 170, 183, 186, 189, 192, 205, 208, 211, 214, 227, 230, 233, 236, 249, 252, 255, 258, 271, 274, 277, 280, 293, 296, 299
Offset: 1

Views

Author

Clark Kimberling, Aug 23 2019

Keywords

Comments

The sequences A026317, A327136, A327137 partition the nonnegative integers.

Examples

			(sin 2, sin 4, ...) = (0.9, -0.7, -0.2, 0.9, -0.5, ...) approximately, so that the differences, in sign, are - + + -  + + - - + - - + ..., with "+" in places 2,3,5,6,... (A026317), "- +" starting in places 1,4,8,11,... (A327136), and "- - +" starting in places 7,10,13,16,... (A327137).

Links

Crossrefs

Cf. A026309, A246303, A026317.

Programs

  • Mathematica
    z = 500; f[x_] := f[x] = Sin[2 x]; t = Range[1, z];
Select[t, f[#] < f[# + 1] &]    (* A026317 *)
Select[t, f[#] > f[# + 1] < f[# + 2] &]  (* A327136 *)
Select[t, f[#] > f[# + 1] > f[# + 2] < f[# + 3] &]   (* A327137 *)

A327140 Numbers k such that cos(2k) > cos(2k+2) > cos(2k+4) < cos(2k+6).

Original entry on oeis.org

3, 6, 9, 22, 25, 28, 31, 44, 47, 50, 53, 66, 69, 72, 75, 88, 91, 94, 97, 110, 113, 116, 119, 132, 135, 138, 141, 154, 157, 160, 163, 179, 182, 185, 188, 201, 204, 207, 210, 223, 226, 229, 232, 245, 248, 251, 254, 267, 270, 273, 276, 289, 292, 295, 298, 311
Offset: 1

Views

Author

Clark Kimberling, Aug 23 2019

Keywords

Comments

The sequences A327138, A327139, A327140 partition the positive integers.

Examples

			(cos 2, cos 4, ...) = (-0.4, -0.6, 0.9, -0.1, -0.8, ...) approximately, so that the differences, in sign, are - + - - + - - + - - + +, with "+" in places 2,5,8,11,12, ... (A327138), "- +" starting in places 1,4,7,10,13,... (A327139), and "- - +" starting in places 3,6,9,22,25,... (A327140).

Links

Crossrefs

Cf. A026309, A246303, A026317, A327138.

Programs

  • Mathematica
    z = 500; f[x_] := f[x] = Cos[2 x]; t = Range[1, z];
Select[t, f[#] < f[# + 1] &]    (* A327138 *)
Select[t, f[#] > f[# + 1] < f[# + 2] &]  (* A327139 *)
Select[t, f[#] > f[# + 1] > f[# + 2] < f[# + 3] &]   (* A327140 *)

A369273 Nonnegative numbers k satisfying sin(k) < sin(k+1) < sin(k+2).

Original entry on oeis.org

0, 5, 6, 11, 12, 17, 18, 24, 25, 30, 31, 36, 37, 42, 43, 44, 49, 50, 55, 56, 61, 62, 68, 69, 74, 75, 80, 81, 86, 87, 88, 93, 94, 99, 100, 105, 106, 112, 113, 118, 119, 124, 125, 130, 131, 132, 137, 138, 143, 144, 149, 150, 156, 157, 162, 163, 168, 169, 174, 175, 181, 182, 187, 188, 193, 194
Offset: 1

Views

Author

R. J. Mathar, Jan 18 2024

Keywords

Comments

Subsequence of terms in A026309: The smaller of two consecutive terms there.
The terms A246389(.)-2 almost match the terms here; the exceptions are terms 178, 220, 266, 310, 555, 599, ... in A246389 where the associates 176, 218, 264, 308, ... are not in this sequence. - Hugo Pfoertner, Jan 18 2024

Crossrefs

Cf. A026309, A369274, A369275.

Programs

  • Maple
    isA369273 := proc(k)
    local s ;
    s := [seq(evalf(sin(k+i)),i=0..2)] ;
    if s[1] < s[2] and s[2] < s[3] then
        true;
    else
        false;
    end if;
end proc:
A369273 := proc(n)
    option remember ;
    if n = 1 then
        0;
    else
        for a from procname(n-1)+1 do
            if isA369273(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A369273(n),n=1..100) ;
  • Mathematica
    Select[Range[0,194], Sin[#]James C. McMahon, Jan 22 2024 *)

A369274 Nonnegative numbers k satisfying sin(k) < sin(k+1) < sin(k+2) < sin(k+3).

Original entry on oeis.org

5, 11, 17, 24, 30, 36, 42, 43, 49, 55, 61, 68, 74, 80, 86, 87, 93, 99, 105, 112, 118, 124, 130, 131, 137, 143, 149, 156, 162, 168, 174, 181, 187, 193, 199, 200, 206, 212, 218, 225, 231, 237, 243, 244, 250, 256, 262, 269, 275, 281, 287, 288, 294, 300, 306, 313, 319, 325, 331, 332, 338, 344, 350
Offset: 1

Views

Author

R. J. Mathar, Jan 18 2024

Keywords

Comments

Subsequence of terms in A026309: The smaller of three consecutive terms there.
Also a subsequence of terms in A369273: The smaller of two consecutive terms there.

Links

Crossrefs

Cf. A026309, A369273, A369275.

Programs

  • Mathematica
    Select[Range[500], Less @@ Sin[Range[#, #+3]] &] (* Paolo Xausa, Mar 19 2024 *)

A369275 Nonnegative numbers k satisfying sin(k) < sin(k+1) < sin(k+2) < sin(k+3) < sin(k+4).

Original entry on oeis.org

42, 86, 130, 199, 243, 287, 331, 375, 419, 463, 507, 532, 576, 620, 664, 708, 752, 796, 840, 909, 953, 997, 1041, 1085, 1129, 1173, 1217, 1242, 1286, 1330, 1374, 1418, 1462, 1506, 1550, 1619, 1663, 1707, 1751, 1795, 1839, 1883, 1927, 1952, 1996, 2040, 2084, 2128, 2172, 2216, 2260, 2329
Offset: 1

Views

Author

R. J. Mathar, Jan 18 2024

Keywords

Comments

This is a subsequence of terms in A369274: The smaller of two consecutive terms there.
Conjecture: there is no followup sequence with sin(k) < sin(k+1) < ... < sin(k+5), i.e., there are no two consecutive integers in this sequence.
It appears that, starting at n=4, a(n) mod 2 has a period of 16 with a periodic part consisting of 8 ones followed by 8 zeros. Similarly, starting at n=4, a(n) mod 4 has a period of 32 with a periodic part consisting of 8 threes followed by 8 zeros followed by 8 ones followed by 8 twos. Tested to 5688 terms. - Gary Detlefs, Jan 20 2024
Both patterns break. The first at n = 7035 and the second at n = 7019. In general this sequence is not periodic mod m for m > 1 as the (period of sin(x))/(1) is irrational. - David A. Corneth, Dec 10 2024
Mathar’s conjecture is true: these are just the numbers which are between Pi*3/2 - 1/2 and Pi*5/2 - 7/2 mod 2*Pi, and reducing the upper bound by 1 leaves it empty. - Charles R Greathouse IV, Dec 10 2024

Links

Crossrefs

Subsequence of A369274 and hence of A369273 and A026309.
Cf. A096444.

Programs

  • Mathematica
    Select[Range[0,2350], Sin[#]James C. McMahon, Jan 20 2024 *)
Flatten[Position[Partition[Sin[Range[2500]],5,1],?(Min[Differences[#]]>0&)]]//Quiet (* _Harvey P. Dale, Dec 10 2024 *)
  • PARI
    first(n) = {
	my(res = List(), streak = 1, s = sin(1));
	for(i = 2, oo,
		c = sin(i);
		if(c > s,
			streak++;
			if(streak >= 5,
				listput(res, i-4);
				if(#res >= n,
					return(res)));
		,
			streak = 1);
s = c); res} \\ David A. Corneth, Dec 11 2024

Formula

a(n) ~ k*n where k = 2*Pi/(Pi - 3) = 44.375... by the Equidistribution Theorem. - Charles R Greathouse IV, Dec 10 2024
Previous Showing 11-16 of 16 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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