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.

Showing 1-10 of 10 results.

A259724 Numbers k such that [r[s*k]] < [s[r*k]], where r = sqrt(2), s=sqrt(3), and [ ] = floor.

Original entry on oeis.org

5, 8, 15, 29, 34, 39, 42, 45, 46, 49, 56, 58, 68, 71, 75, 87, 92, 95, 99, 102, 105, 109, 112, 116, 121, 124, 127, 128, 131, 145, 150, 157, 162, 169, 174, 177, 184, 187, 191, 198, 203, 206, 213, 232, 237, 240, 243, 244, 247, 254, 256, 266, 269, 273, 285, 290
Offset: 1

Views

Author

Clark Kimberling, Jul 15 2015

Keywords

Comments

Suppose that r and s are distinct real numbers, and let f(r,s,k) = [s[r*k]] - [r[s*k]]. Let (G(n)) be the sequence of those k for which f(r,s,k) > 0, let (E(n)) be those for which f(r,s,k) = 0, and (L(n)), those for which f(r,s,k) < 0. Clearly (G(n), E(n), L(n)) partition the positive integers. In particular, A259724, A259725, A259726 partition the positive integers.
Conjecture: the limits g = lim G(n)/n, e = lim E(n)/n, el = lim L(n)/n exist; if so, then 1/g + 1/e + 1/el = 1.

Links

Crossrefs

Cf. A259725, A259746.

Programs

  • Mathematica
    z = 1000; r = Sqrt[2]; s = Sqrt[3];
u = Table[Floor[r*Floor[s*n]], {n, 1, z}];
v = Table[Floor[s*Floor[r*n]], {n, 1, z}];
Select[Range[400], u[[#]] < v[[#]] &]  (* A259724 *)
Select[Range[200], u[[#]] == v[[#]] &] (* A259725 *)
Select[Range[200], u[[#]] > v[[#]] &]  (* A259726 *)

A259725 Numbers k such that [r[s*k]] = [s[r*k]], where r = sqrt(2), s=sqrt(3), and [ ] = floor.

Original entry on oeis.org

1, 4, 10, 13, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 27, 28, 30, 31, 32, 35, 37, 38, 44, 47, 50, 51, 53, 54, 57, 60, 61, 63, 64, 66, 69, 73, 76, 78, 79, 80, 81, 83, 85, 86, 88, 90, 97, 98, 100, 103, 104, 106, 107, 110, 113, 114, 117, 120, 126, 129, 132, 133
Offset: 1

Views

Author

Clark Kimberling, Jul 15 2015

Keywords

Comments

Suppose that r and s are distinct real numbers, and let f(r,s,k) = [s[r*k]] - [r[s*k]]. Let (G(n)) be the sequence of those k for which f(r,s,k) > 0, let (E(n)) be those for which f(r,s,k) = 0, and (L(n)), those for which f(r,s,k) < 0. Clearly (G(n), E(n), L(n)) partition the positive integers. In particular, A259724, A259725, A259726 partition the positive integers.
Conjecture: the limits g = lim G(n)/n, e = lim E(n)/n, el = lim L(n)/n exist; if so, then 1/g + 1/e + 1/el = 1.

Links

Crossrefs

Cf. A259724, A259746.

Programs

  • Mathematica
    z = 1000; r = Sqrt[2]; s = Sqrt[3];
u = Table[Floor[r*Floor[s*n]], {n, 1, z}];
v = Table[Floor[s*Floor[r*n]], {n, 1, z}];
Select[Range[400], u[[#]] < v[[#]] &]   (* A259724 *)
Select[Range[200], u[[#]] == v[[#]] &]  (* A259725 *)
Select[Range[200], u[[#]] > v[[#]] &]   (* A259726 *)

A259745 Triangular numbers representable as x*y+x+y such that x and y are triangular numbers, x>=y>0.

Original entry on oeis.org

3, 15, 21, 91, 120, 153, 351, 406, 703, 741, 1891, 3081, 3403, 4465, 5151, 5565, 6555, 9453, 9591, 18721, 22791, 23871, 38226, 39903, 46056, 52326, 79401, 85491, 91378, 104653, 159895, 187578, 207690, 222111, 227475, 229503, 266815, 274911, 280875, 326028, 334971, 354903
Offset: 1

Views

Author

Alex Ratushnyak, Jul 04 2015

Keywords

Comments

A subsequence of A000217.
Up to 10^12, only 4 terms admit two representations, namely 3081, 22791, 410871 and 1675365. - Giovanni Resta, Jul 19 2015

Examples

			a(1) = 3 = 1*1 + 1 + 1.
a(2) = 15 = 3*3 + 3 + 3.
a(3) = 21 = 10*1 + 10 + 1.
a(4) = 91 = 45*1 + 45 + 1.
a(5) = 120 = 10*10 + 10 + 10.
a(6) = 153 = 6*21 + 6 + 21.

Links

Crossrefs

Cf. A000217, A259746.

A259584 Numbers k such that [r[s*k]] - [s[r*k]] = -2, where r = sqrt(2), s=sqrt(3), and [ ] = floor.

Original entry on oeis.org

116, 314, 512, 657, 1340, 1422, 1620, 1818, 1900, 2161, 2243, 2441, 2639, 2982, 3124, 3322, 3747, 3800, 3945, 4027, 4143, 4225, 4766, 5251, 5449, 5531, 5729, 5927, 6125, 6270, 6352, 6953, 7091, 7233, 7431, 7711, 7774, 7856, 8054, 8252, 8457, 8595, 9278, 9360
Offset: 1

Views

Author

Clark Kimberling, Jul 15 2015

Keywords

Comments

It is easy to prove that [r[s*k]] - [s[r*k]] ranges from -2 to 2. For k = 1 to 10, the values of [r[s*k]] - [s[r*k]] are 0, 1, 1, 0, -1, 1, 1, -1, 1, 0.
The first -2 occurs when k = 116.

Links

Crossrefs

Cf. A259585, A259586, A259587, A259724, A259725, A259746.

Programs

  • Mathematica
    z = 12000; r = Sqrt[2]; s = Sqrt[3];
u = Table[Floor[r*Floor[s*n]], {n, 1, z}];
v = Table[Floor[s*Floor[r*n]], {n, 1, z}];
Flatten[Position[u - v, -2]] (* A259584 *)
Take[Flatten[Position[u - v, -1]], 100] (* A259585 *)
Take[Flatten[Position[u - v, 0]], 100]  (* A259725 *)
Take[Flatten[Position[u - v, 1]], 100]  (* A259587 *)
Take[Flatten[Position[u - v, 2]], 100]  (* A259586 *)
Select[Range[10000],Floor[Sqrt[2]Floor[Sqrt[3]#]]-Floor[Sqrt[3]Floor[ Sqrt[ 2]#]]==-2&] (* Harvey P. Dale, Dec 01 2016 *)

A259585 Numbers k such that [r[s*k]] - [s[r*k]] = -1, where r = sqrt(2), s=sqrt(3), and [ ] = floor.

Original entry on oeis.org

5, 8, 15, 29, 34, 39, 42, 45, 46, 49, 56, 58, 68, 71, 75, 87, 92, 95, 99, 102, 105, 109, 112, 121, 124, 127, 128, 131, 145, 150, 157, 162, 169, 174, 177, 184, 187, 191, 198, 203, 206, 213, 232, 237, 240, 243, 244, 247, 254, 256, 266, 269, 273, 285, 290, 295
Offset: 1

Views

Author

Clark Kimberling, Jul 15 2015

Keywords

Comments

It is easy to prove that [r[s*k]] - [s[r*k]] ranges from -2 to 2.

Examples

			For k = 1 to 10, the values of [r[s*k]] - [s[r*k]] are 0, 1, 1, 0, -1, 1, 1, -1, 1, 0, so that a(1) = 5.

Links

Crossrefs

Cf. A259584, A259586, A259587, A259724, A259725, A259746.

Programs

  • Mathematica
    z = 12000; r = Sqrt[2]; s = Sqrt[3];
u = Table[Floor[r*Floor[s*n]], {n, 1, z}];
v = Table[Floor[s*Floor[r*n]], {n, 1, z}];
Flatten[Position[u - v, -2]] (* A259584 *)
Take[Flatten[Position[u - v, -1]], 100] (* A259585 *)
Take[Flatten[Position[u - v, 0]], 100]  (* A259725 *)
Take[Flatten[Position[u - v, 1]], 100]  (* A259587 *)
Take[Flatten[Position[u - v, 2]], 100]  (* A259586 *)

A259586 Numbers k such that [r[s*k]] - [s[r*k]] = 2, where r = sqrt(2), s=sqrt(3), and [ ] = floor.

Original entry on oeis.org

41, 67, 70, 123, 130, 205, 212, 328, 350, 403, 410, 444, 526, 548, 555, 608, 671, 700, 724, 750, 753, 806, 869, 888, 898, 951, 1026, 1033, 1067, 1086, 1096, 1149, 1224, 1231, 1265, 1291, 1294, 1347, 1376, 1429, 1489, 1504, 1545, 1571, 1574, 1627, 1709, 1716
Offset: 1

Views

Author

Clark Kimberling, Jul 15 2015

Keywords

Comments

It is easy to prove that [r[s*k]] - [s[r*k]] ranges from -2 to 2. For k = 1 to 10, the values of [r[s*k]] - [s[r*k]] are 0, 1, 1, 0, -1, 1, 1, -1, 1, 0; the first appearance of 2 is when k = 41.

Links

Crossrefs

Cf. A259584, A259585, A259587, A259724, A259725, A259746.

Programs

  • Mathematica
    z = 12000; r = Sqrt[2]; s = Sqrt[3];
u = Table[Floor[r*Floor[s*n]], {n, 1, z}];
v = Table[Floor[s*Floor[r*n]], {n, 1, z}];
Flatten[Position[u - v, -2]] (* A259584 *)
Take[Flatten[Position[u - v, -1]], 100] (* A259585 *)
Take[Flatten[Position[u - v, 0]], 100]  (* A259725 *)
Take[Flatten[Position[u - v, 1]], 100]  (* A259587 *)
Take[Flatten[Position[u - v, 2]], 100]  (* A259586 *)
Select[Range[2000],Floor[Sqrt[2]Floor[Sqrt[3]#]]-Floor[Sqrt[3]Floor[Sqrt[2]#]]==2&] (* Harvey P. Dale, Aug 10 2024 *)

A259587 Numbers k such that [r[s*k]] - [s[r*k]] = 2, where r = sqrt(2), s=sqrt(3), and [ ] = floor.

Original entry on oeis.org

2, 3, 6, 7, 9, 11, 12, 14, 26, 33, 36, 40, 43, 48, 52, 55, 59, 62, 65, 72, 74, 77, 82, 84, 89, 91, 93, 94, 96, 101, 108, 111, 115, 118, 119, 122, 125, 134, 137, 140, 141, 144, 147, 148, 149, 151, 152, 154, 159, 164, 171, 175, 178, 181, 188, 190, 193, 194
Offset: 1

Views

Author

Clark Kimberling, Jul 15 2015

Keywords

Comments

It is easy to prove that [r[s*k]] - [s[r*k]] ranges from -2 to 2. For k = 1 to 10, the values of [r[s*k]] - [s[r*k]] are 0, 1, 1, 0, -1, 1, 1, -1, 1, 0; the first appearance of 2 is when k = 41.

Links

Crossrefs

Cf. A259584, A259585, A259586, A259724, A259725, A259746.

Programs

  • Mathematica
    z = 12000; r = Sqrt[2]; s = Sqrt[3];
u = Table[Floor[r*Floor[s*n]], {n, 1, z}];
v = Table[Floor[s*Floor[r*n]], {n, 1, z}];
Flatten[Position[u - v, -2]] (* A259584 *)
Take[Flatten[Position[u - v, -1]], 100] (* A259585 *)
Take[Flatten[Position[u - v, 0]], 100]  (* A259725 *)
Take[Flatten[Position[u - v, 1]], 100]  (* A259587 *)
Take[Flatten[Position[u - v, 2]], 100]  (* A259586 *)

A262242 Triangular numbers representable as 2^x + 2^y.

Original entry on oeis.org

3, 6, 10, 36, 66, 136, 528, 2080, 8256, 32896, 131328, 524800, 2098176, 8390656, 33558528, 134225920, 536887296, 2147516416, 8590000128, 34359869440, 137439215616, 549756338176, 2199024304128, 8796095119360, 35184376283136, 140737496743936, 562949970198528
Offset: 1

Views

Author

Alex Ratushnyak, Sep 15 2015

Keywords

Links

Crossrefs

Cf. A000217, A225390, A226499, A227027, A259745, A259746.

Programs

  • Python
    def isTriangular(a):
    sr = 1 << (int.bit_length(a) >> 1)
    a += a
    while a < sr*(sr+1):  sr>>=1
    b = sr>>1
    while b:
      s = sr+b
      if a >= s*(s+1):  sr = s
      b>>=1
    return (a==sr*(sr+1))
for a in range(1,200):
    for b in range(a):
        c = (1<

Formula

Conjectures from Colin Barker, Sep 16 2015: (Start)
a(n) = 2^(n-5)*(2^n+4) for n>5.
a(n) = 6*a(n-1)-8*a(n-2) for n>7.
G.f.: x*(240*x^6+28*x^5-70*x^4+24*x^3-2*x^2-12*x+3) / ((2*x-1)*(4*x-1)).
(End)

A262251 Triangular numbers representable as 2^x + 3^y.

Original entry on oeis.org

3, 10, 28, 91
Offset: 1

Views

Author

Alex Ratushnyak, Sep 16 2015

Keywords

Comments

No other terms such that 0 <= x,y < 2000.
No other terms such that 0 <= x,y < 5250. - Michael S. Branicky, Mar 10 2021

Examples

			a(1) = 3 = 2^1 + 3^0.
a(4) = 91 = 2^6 + 3^3.

Crossrefs

Cf. A225390, A226499, A227027, A259745, A259746, A262242.
Intersection of A000217 and A004050.

Programs

  • PARI
    isok(t) = {for (k=0, logint(t, 2), my(tt = t - 2^k); if (tt, p = valuation(tt, 3); if (tt == 3^p, return(1))););}
lista(nn) = for (n=1, nn, if (isok(t=n*(n+1)/2), print1(t, ", "))); \\ Michel Marcus, Sep 20 2015
  • PARI
    select(x->ispolygonal(x, 3), setbinop(f, [0..20], [0..20])) \\ Michel Marcus, Mar 10 2021
  • Python
    from sympy import integer_nthroot
def auptoexponent(maxexp):
  sums = set(2**x + 3**y for x in range(maxexp) for y in range(maxexp))
  iroots = set(integer_nthroot(2*s, 2)[0] for s in sums)
  return sorted(set(r*(r+1)//2 for r in iroots if r*(r+1)//2 in sums))
print(auptoexponent(500)) # Michael S. Branicky, Mar 10 2021

A262724 Triangular numbers representable as 3^x + y^3.

Original entry on oeis.org

1, 3, 10, 28, 36, 91, 1081, 2278, 2926, 8001, 46665, 5639761, 10911456, 166066200, 341532180, 3137785371, 1647882316985625, 875366737297292691171, 465198187808352499674075441
Offset: 1

Views

Author

Alex Ratushnyak, Sep 28 2015

Keywords

Comments

Any further terms are greater than 10^30. - Charles R Greathouse IV, Oct 02 2015

Crossrefs

Cf. A000217, A225390, A226499, A227027, A259745, A259746, A262242, A262251.

Programs

  • PARI
    list(lim)=my(v=List(),X,t); for(x=0,logint(lim\=1,3), X=3^x; for(y=0, sqrtnint(lim-X,3), if(ispolygonal(t=X+y^3,3), listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Sep 28 2015

Extensions

a(17)-a(19) from Charles R Greathouse IV, Sep 28 2015
Showing 1-10 of 10 results.

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

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