A277723 -id:A277723 - cp's OEIS frontend

A277725 Intersection of A158919 and A277723.

0, 12, 18, 31, 49, 62, 68, 80, 93, 99, 112, 130, 136, 143, 161, 174, 180, 211, 217, 224, 242, 248, 255, 261, 286, 292, 323, 329, 336, 342, 354, 360, 367, 373, 404, 410, 423, 435, 441, 448, 454, 472, 485, 491, 516, 522, 535, 553, 560, 566, 572, 584, 597, 603, 616, 634, 640, 647, 665, 678, 684, 709, 715

Offset: 1

N. J. A. Sloane, Oct 30 2016

nonn

See A277728 for discussion.

Cf. A003144, A003145, A003146, A058265, A158919, A275926, A276799, A276800, A277722, A277723, A277724, A277725, A277726, A277727, A277728.

A277726 Intersection of A277722 and A277723.

Original entry on oeis.org

0, 6, 37, 43, 74, 87, 118, 155, 186, 192, 199, 230, 236, 267, 280, 304, 311, 317, 348, 385, 392, 416, 429, 460, 466, 497, 504, 510, 541, 578, 622, 659, 690, 696, 703, 734, 740, 771, 784, 808, 815, 852, 889, 896, 920, 933, 964, 970, 1001, 1008, 1014, 1045, 1082, 1126, 1163, 1194, 1200, 1207, 1238, 1244, 1275, 1288, 1312, 1319, 1356, 1387, 1393, 1400, 1424

Offset: 1

N. J. A. Sloane, Oct 30 2016

nonn

See A277728 for discussion.

Cf. A003144, A003145, A003146, A058265, A158919, A275926, A276799, A276800, A277722, A277723, A277724, A277725, A277726, A277727, A277728.

Digits := 120;
isA277722 := proc(n)
    a276800 :=  3.3829757679062374941227085364550345869493820437485761820195626772353718960099402922235933340043661396041006 ;
    for x from floor((n-3)/a276800) to (n+3)/a276800 do
        if floor(x*a276800) = n then
            return true;
        end if;
    end do:
    return false;
end proc:
isA277723 := proc(n)
    a276801 :=  6.2222625231203986266745611011083211873735607898461684287983213166395751180919067179620287534326731537460804;
    for x from floor((n-3)/a276801) to (n+3)/a276801 do
        if floor(x*a276801) = n then
            return true;
        end if;
    end do:
    return false;
end proc:
isA277726 := proc(n)
    isA277722(n) and isA277723(n) ;
end proc:
for n from 0 to 8000 do
    if isA277726(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Nov 02 2016

Corrected by R. J. Mathar, Nov 01 2016

A277727 Union of A158919, A277722, A277723.

Original entry on oeis.org

0, 1, 3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 16, 18, 20, 22, 23, 24, 25, 27, 29, 30, 31, 33, 34, 36, 37, 38, 40, 42, 43, 44, 45, 47, 49, 50, 51, 53, 54, 55, 56, 57, 58, 60, 62, 64, 66, 67, 68, 69, 71, 73, 74, 75, 77, 79, 80, 81, 82, 84, 86, 87, 88, 90, 91, 93, 94, 95, 97, 98, 99, 101, 103, 104, 105, 106

Offset: 1

N. J. A. Sloane, Oct 30 2016

nonn

See A277728 for discussion.

Cf. A003144, A003145, A003146, A058265, A158919, A275926, A276799, A276800, A277722, A277723, A277724, A277725, A277726, A277727, A277728.

A277728 Numbers not in any of A158919, A277722, A277723.

Original entry on oeis.org

2, 4, 8, 15, 17, 19, 21, 26, 28, 32, 35, 39, 41, 46, 48, 52, 59, 61, 63, 65, 70, 72, 76, 78, 83, 85, 89, 92, 96, 100, 102, 107, 109, 113, 116, 120, 122, 127, 129, 133, 140, 144, 146, 151, 153, 157, 159, 164, 166, 170, 173, 177, 181, 184, 188, 190, 195, 197, 201, 203, 208, 210, 212, 214, 221

Offset: 1

N. J. A. Sloane, Oct 30 2016

nonn

Let tau be the tribonacci constant (A058265). Although 1/tau + 1/tau^2 + 1/tau^3 = 1, by Uspensky's 1927 theorem, the three sequences floor(n*tau) (A158919), floor(n*tau^2) (A277722), and floor(n*tau^3) (A277723) cannot form a partition of the nonnegative integers. (Compare Beatty's theorem.)
Entries A277724-A277727 investigate how these three sequences meet, and the present sequence gives the numbers not in any of the three sequences. Any two of the three sequences have a nontrivial intersection, while the intersection of all three is {0}.
On the other hand, the three sequences A003144, A003145, A003146, which arise from the tribonacci word, DO form a partition of the positive integers and are closely connected with the three sequences mentioned in the definition.
It would be nice to have b-files for all the sequences mentioned here. (Many do, but some do not.)

Jean-François Alcover, Table of n, a(n) for n = 1..2955
S. Beatty, A. Ostrowski, J. Hyslop, and A. C. Aitken, Problems and Solutions: Solutions: 3177, Amer. Math. Monthly, 34 (1927), pp. 159-160.
R. L. Graham, On a theorem of Uspensky, Amer. Math. Mnthly, 70 (1963): 407-409.
A. J. Hildebrand, Junxian Li, Xiaomin Li, Yun Xie, Almost Beatty Partitions, arXiv:1809.08690 [math.NT], 2018.
J. V. Uspensky, On a problem arising out of the theory of a certain game, Amer. Math. Mnthly., 34 (1927), 516-521.

Cf. A003144, A003145, A003146, A058265, A158919, A184820, A277722, A277723, A277724, A277725, A277726, A277727.

maxTerm = 10000;
a19[n_] := Floor[n*Root[#^3 - #^2 - # - 1&, 1]];
a22[n_] := Floor[n*Root[#^3 - 3 #^2 - # - 1&, 1]];
a23[n_] := Floor[n*Root[#^3 - 7 #^2 + 5 # - 1&, 1]];
A19 = Reap[k = 1; While[a19[k] <= maxTerm, Sow[a19[k++]]]][[2, 1]];
A22 = Reap[k = 1; While[a22[k] <= maxTerm, Sow[a22[k++]]]][[2, 1]];
A23 = Reap[k = 1; While[a23[k] <= maxTerm, Sow[a23[k++]]]][[2, 1]];
Select[Range[maxTerm], FreeQ[A19, #] && FreeQ[A22, #] && FreeQ[A23, #]&] (* Jean-François Alcover, Dec 06 2018 *)

A158919 Beatty sequence for the tribonacci constant tau (A058265): a(n) = floor(n*tau).

Original entry on oeis.org

0, 1, 3, 5, 7, 9, 11, 12, 14, 16, 18, 20, 22, 23, 25, 27, 29, 31, 33, 34, 36, 38, 40, 42, 44, 45, 47, 49, 51, 53, 55, 57, 58, 60, 62, 64, 66, 68, 69, 71, 73, 75, 77, 79, 80, 82, 84, 86, 88, 90, 91, 93, 95, 97, 99, 101, 103, 104, 106, 108, 110

Offset: 0

Eric Culver (weux082690(AT)yahoo.com), Mar 30 2009

nonn

Also called the spectrum of tau. Note that A276384 agrees with this sequence for n <= 17160 but disagrees beyond that point. In fact a(17161) = 31564, whereas A276384(17161) = 31563. - N. J. A. Sloane, Sep 03 2016

			a(3) = floor(3*q) = floor(3*1.8392867...) = floor(5.51786...) = 5.

N. J. A. Sloane, Table of n, a(n) for n = 0..20000
A. J. Hildebrand, Junxian Li, Xiaomin Li, Yun Xie, Almost Beatty Partitions, arXiv:1809.08690 [math.NT], 2018.
Wolfdieter Lang, The Tribonacci and ABC Representations of Numbers are Equivalent, arXiv preprint arXiv:1810.09787 [math.NT], 2018.
N. J. A. Sloane, Table of n, a(n) for n = 0..100000
Index entries for sequences related to Beatty sequences

Cf. A058265, A140099 (spectrum of 1+tau), A276384, A277722, A277723.

[Floor(n * (1/3 + 1/3 * (19 - 3 * Sqrt(33))^(1/3) + 1/3 * (19 + 3 * Sqrt(33))^(1/3))) : n in [0..80]]; // Vincenzo Librandi, Oct 28 2018

x := (1/3)*(1+(19+3*sqrt(33))^(1/3)+(19-3*sqrt(33))^(1/3)) ;
[seq(floor(n*x),n=0..200)]; # R. J. Mathar, Sep 11 2011

a[n_] := Floor[n Root[#^3 - #^2 - # - 1&, 1]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Oct 28 2018 *)

a(n) = floor(n*A058265).

A277722 a(n) = floor(n*tau^2) where tau is the tribonacci constant (A058265).

Original entry on oeis.org

0, 3, 6, 10, 13, 16, 20, 23, 27, 30, 33, 37, 40, 43, 47, 50, 54, 57, 60, 64, 67, 71, 74, 77, 81, 84, 87, 91, 94, 98, 101, 104, 108, 111, 115, 118, 121, 125, 128, 131, 135, 138, 142, 145, 148, 152, 155, 158, 162, 165, 169, 172, 175, 179, 182, 186, 189, 192, 196, 199, 202, 206, 209, 213, 216, 219

Offset: 0

N. J. A. Sloane, Oct 30 2016

nonn

JungHwan Min, Table of n, a(n) for n = 0..10000
A. J. Hildebrand, Junxian Li, Xiaomin Li and Yun Xie, Almost Beatty Partitions, arXiv:1809.08690 [math.NT], 2018.

Cf. A058265, A158919, A276800, A277723.

A277722 := proc(n)
    a276800 :=  3.3829757679062374941227085364550345869493820437485761820195626772353718960099402922235933340043661396041006 ;
    floor(n*a276800) ;
end proc:
seq(A277722(n),n=0..100) ; # R. J. Mathar, Nov 02 2016

A277722[n_] := Floor[n (1/3 (1 + (19 - 3 Sqrt[33])^(1/3) + (19 + 3 Sqrt[33])^(1/3)))^2]; Array[A277722, 66, 0] (* JungHwan Min, Nov 06 2016 *)

A277724 Intersection of A158919 and A277722.

Original entry on oeis.org

0, 3, 16, 20, 23, 27, 33, 40, 47, 57, 60, 64, 71, 77, 84, 91, 101, 104, 108, 115, 121, 125, 128, 145, 148, 152, 158, 165, 169, 172, 182, 189, 196, 202, 206, 209, 213, 226, 233, 240, 246, 250, 253, 257, 263, 270, 274, 277, 290, 294, 297, 301, 307, 314, 321, 331, 334, 338, 345, 351, 358, 375, 378, 382

Offset: 1

N. J. A. Sloane, Oct 30 2016

nonn

See A277728 for discussion.

Cf. A003144, A003145, A003146, A058265, A158919, A275926, A276799, A276800, A277722, A277723, A277724, A277725, A277726, A277727, A277728.

cp's OEIS Frontend

A277725 Intersection of A158919 and A277723.

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A277726 Intersection of A277722 and A277723.

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A277727 Union of A158919, A277722, A277723.

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A277728 Numbers not in any of A158919, A277722, A277723.

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A158919 Beatty sequence for the tribonacci constant tau (A058265): a(n) = floor(n*tau).

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A277722 a(n) = floor(n*tau^2) where tau is the tribonacci constant (A058265).

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A277724 Intersection of A158919 and A277722.

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