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 46 results. Next

A184912 n+[ns/r]+[nt/r]+[nu/r], where []=floor and r=2^(1/5), s=r^2, t=r^3, u=r^4.

Original entry on oeis.org

4, 9, 13, 19, 23, 28, 34, 39, 43, 49, 53, 58, 63, 69, 73, 79, 83, 88, 93, 98, 103, 109, 113, 118, 122, 128, 133, 138, 143, 148, 152, 158, 163, 168, 174, 178, 183, 188, 193, 197, 204, 208, 213, 218, 223, 227, 233, 238, 243, 247, 253, 257, 262, 268, 273, 277, 283, 287, 292, 297, 303, 307, 313, 318, 322, 328, 332, 338, 343, 348, 352, 358, 362, 368, 372, 378, 382, 387, 392, 397, 402, 408, 412, 417, 422, 427, 431, 438, 442, 447, 452, 457, 461, 467, 472, 477, 482, 487, 492, 496, 503, 507, 512, 517, 522, 526, 532, 537, 542, 547, 552, 556, 562, 566, 572, 577, 582, 586, 592, 596
Offset: 1

Views

Author

Clark Kimberling, Jan 25 2011

Keywords

Comments

The sequences A184912-A184915 partition the positive integers:
A184912: 4,9,13,19,23,28,34,...
A184913: 3,7,11,16,20,24,30,...
A184914: 2,6,10,14,17,21,26,...
A184915: 1,5,8,12,15,18,22,...
The joint ranking method of A184812 is extended here to four numbers r,s,t,u, as follows: jointly rank the sets {h*r}, {i*s}, {j*t}, {k*u}, h>=1, i>=1, j>=1, k>=1.
The position of n*r in the joint ranking is
n+[sn/r]+[tn/r]+[un/r], and likewise for the
positions of n*s, n*t, and n*u.

Crossrefs

Cf. A184312, A184913, A184914, A184915.

Programs

  • Mathematica
    r=2^(1/5); s=2^(2/5); t=2^(3/5); u=2^(4/5);
a[n_]:=n+Floor[n*s/r]+Floor[n*t/r]+Floor[n*u/r];
b[n_]:=n+Floor[n*r/s]+Floor[n*t/s]+Floor[n*u/s];
c[n_]:=n+Floor[n*r/t]+Floor[n*s/t]+Floor[n*u/t];
d[n_]:=n+Floor[n*r/u]+Floor[n*s/u]+Floor[n*t/u];
Table[a[n],{n,1,120}]  (* A184912 *)
Table[b[n],{n,1,120}]  (* A184913 *)
Table[c[n],{n,1,120}]  (* A184914 *)
Table[d[n],{n,1,120}]  (* A184915 *)

Formula

a(n)=n+[ns/r]+[nt/r]+[nu/r], where []=floor and
r=2^(1/5), s=r^2, t=r^3, u=r^4.

A378142 a(n) = n + floor(n*s/r) + floor(n*t/r), where r=2^(1/4), s=2^(1/2), t=2^(3/4).

Original entry on oeis.org

3, 6, 10, 13, 17, 21, 24, 28, 31, 35, 39, 42, 46, 49, 53, 57, 61, 64, 67, 71, 74, 79, 82, 85, 89, 92, 97, 100, 104, 107, 110, 115, 118, 122, 125, 128, 133, 136, 140, 143, 146, 150, 154, 158, 161, 165, 168, 172, 176, 179, 183, 186, 190, 194, 197, 201, 204
Offset: 1

Views

Author

Clark Kimberling, Jan 13 2025

Keywords

Comments

The sequences A378142, A378185, A379510, partition the positive integers (A000027), as proved at A184812:
A378142: 3,6,10,13,17,21,24,28,32,35,...
A378185: 2,5,8,11,14,18,20,23,26,29,,...
A379510: 1,4,7,9,12,15,16,19,22,25,27,...
For each integer k >= 1, write "a" if k=A378142(n) for some n, "b" if k=A378185(n) for some n, and "c" if k=A379510(n) for some n. Concatenating these letters for k = 1,2,3,... spells the following infinite word:
cbacbacbcabcabccabcbacbacbcabcacbcabcbacbacbcacbacbcabcbacbcabcacbacbcabcabcbcacbacbacbcabcabccbacbacb...

Crossrefs

Cf. A000027, A010767, A184812, A378185, A379510.

Programs

  • Mathematica
    r=2^(1/4); s=2^(1/2); t=2^(3/4);
a[n_]:=n+Floor[n*s/r]+Floor[n*t/r];
b[n_]:=n+Floor[n*r/s]+Floor[n*t/s];
c[n_]:=n+Floor[n*r/t]+Floor[n*s/t];
Table[a[n], {n, 1, 120}]  (* A378142 *)
Table[b[n], {n, 1, 120}]  (* A378185 *)
Table[c[n], {n, 1, 120}]  (* A379510 *)

Formula

a(n) = n + [w*n] + [w^2 n], where w = 2^(1/4) and [ ] = floor.

Extensions

Name corrected by Clark Kimberling, Jan 20 2025

A189361 a(n) = n + floor(n*s/r) + floor(n*t/r); r=1, s=sqrt(2), t=sqrt(3).

Original entry on oeis.org

3, 7, 12, 15, 20, 24, 28, 32, 36, 41, 45, 48, 53, 57, 61, 65, 70, 74, 77, 82, 86, 91, 94, 98, 103, 107, 111, 115, 120, 123, 127, 132, 136, 140, 144, 148, 153, 156, 161, 165, 169, 173, 177, 182, 185, 190, 194, 198, 202, 206, 211, 215, 218, 223, 227, 231, 235, 240, 244, 247, 252, 256, 261, 264, 268, 273, 277, 281, 285, 289, 293, 297, 302, 306, 310, 314, 318, 323, 326, 331, 335, 339, 343
Offset: 1

Views

Author

Clark Kimberling, Apr 20 2011

Keywords

Comments

This is one of three sequences that partition the positive integers. In general, suppose that r, s, t are positive real numbers for which the sets {i/r: i>=1}, {j/s: j>=1}, {k/t: k>=1} are pairwise disjoint. Let a(n) be the rank of n/r when all the numbers in the three sets are jointly ranked. Define b(n) and c(n) as the ranks of n/s and n/t. It is easy to prove that
a(n) = n + [n*s/r] + [n*t/r],
b(n) = n + [n*r/s] + [n*t/s],
c(n) = n + [n*r/t] + [n*s/t], where []=floor.
Taking r=1, s=sqrt(2), t=sqrt(3) gives
a=A189361, b=A189362, c=A189363.

Links

Crossrefs

Cf. A189362, A189363, A184812, A189383, A189386, A189395.

Programs

  • Mathematica
    r = 1; s = 2^(1/2); t = 3^(1/2);
a[n_] := n + Floor[n*s/r] + Floor[n*t/r];
b[n_] := n + Floor[n*r/s] + Floor[n*t/s];
c[n_] := n + Floor[n*r/t] + Floor[n*s/t]
Table[a[n], {n, 1, 120}]  (*A189361*)
Table[b[n], {n, 1, 120}]  (*A189362*)
Table[c[n], {n, 1, 120}]  (*A189363*)

A184835 a(n) = n + floor(n/t) + floor(n/t^2) + floor(n/t^3) + floor(n/t^4), where t is the pentanacci constant.

Original entry on oeis.org

1, 3, 4, 7, 8, 10, 11, 15, 16, 18, 19, 22, 23, 25, 27, 31, 32, 34, 35, 38, 39, 41, 43, 46, 47, 49, 50, 53, 54, 57, 60, 62, 63, 65, 67, 69, 70, 73, 75, 77, 78, 80, 82, 84, 86, 89, 91, 93, 94, 96, 98, 100, 101, 104, 106, 108, 109, 112, 114, 116, 119, 121, 123, 124, 126, 128, 130, 131, 134, 136, 138, 139, 141, 143, 146, 148, 150, 152, 154, 155, 157, 159, 161, 163, 165, 167, 169, 170, 173, 175, 177, 179, 182, 183, 185, 186, 189, 190, 193, 194, 197, 198, 200, 201, 205, 206, 209, 210, 213, 214, 216, 217, 220, 222, 224, 227, 228
Offset: 1

Views

Author

Paul D. Hanna, Jan 23 2011

Keywords

Comments

This is one of five sequences that partition the positive integers.
Given t is the pentanacci constant, then the following sequences are disjoint:
. A184835(n) = n + [n/t] + [n/t^2] + [n/t^3] + [n/t^4],
. A184836(n) = n + [n*t] + [n/t] + [n/t^2] + [n/t^3],
. A184837(n) = n + [n*t] + [n*t^2] + [n/t] + [n/t^2],
. A184838(n) = n + [n*t] + [n*t^2] + [n*t^3] + [n/t],
. A184839(n) = n + [n*t] + [n*t^2] + [n*t^3] + [n*t^4], where []=floor.
This is a special case of Clark Kimberling's results given in A184812.

Examples

			Given t = pentanacci constant, then t = 1 + 1/t + 1/t^2 + 1/t^3 + 1/t^4,
t = 1.965948236645..., t^2 = 3.864952469169..., t^3 = 7.598296491482..., t^4 = 14.93785758893..., t^5 = 29.36705478623...

Crossrefs

Cf. A184836, A184837, A184838, A184839; A184820, A184823, A184812, A103814.

Programs

  • Mathematica
    With[{pc=x/.FindRoot[x^5-x^4-x^3-x^2-x-1==0,{x,1.96},WorkingPrecision-> 100]}, Table[n+Total[Table[Floor[n/pc^i],{i,4}]],{n,150}]] (* Harvey P. Dale, Jun 21 2011 *)
  • PARI
    {a(n)=local(t=real(polroots(1+x+x^2+x^3+x^4-x^5)[1])); n+floor(n/t)+floor(n/t^2)+floor(n/t^3)+floor(n/t^4)}

Formula

Limit a(n)/n = t = 1.9659482366454853371899373...
a(n) = n + floor(n*p/u) + floor(n*q/u) + floor(n*r/u) + floor(n*s/u), where p=t, q=t^2, r=t^3, s=t^4, u=t^5, and t is the pentanacci constant.

A184916 n+[sn/r]+[tn/r]+[un/r], where []=floor and r=1, s=2^(1/4), t=s^2, u=s^3.

Original entry on oeis.org

4, 9, 15, 19, 25, 31, 35, 41, 46, 51, 57, 62, 67, 72, 78, 83, 89, 94, 98, 104, 109, 115, 120, 125, 131, 135, 142, 147, 152, 157, 162, 168, 173, 179, 183, 188, 195, 199, 205, 210, 214, 220, 226, 231, 236, 242, 247, 252, 258, 263, 268, 273, 279, 284, 289, 295, 299, 305, 311, 315, 321, 326, 331, 337, 342, 347, 352, 358, 364, 368, 374, 379, 384, 390, 396, 400, 405, 411, 415, 422, 427, 431, 437, 442, 448, 453, 459, 463, 468, 475, 480, 485, 490, 495, 500, 506, 512, 516, 522, 527, 532, 538, 543, 548, 553, 559, 564, 569, 575, 579, 585, 591, 596, 601, 606, 612, 617, 622, 628, 632
Offset: 1

Views

Author

Clark Kimberling, Jan 26 2011

Keywords

Comments

The sequences A184916-A184919 partition the positive integers:
A184916: 4,9,15,19,25,31,35,41,...
A184917: 3,7,12,16,21,26,29,34,...
A184918: 2,6,10,13,17,22,24,28,...
A184919: 1,5,8,11,14,18,20,23,27,...
The joint ranking method of A184812 is extended here to four numbers r,s,t,u, as follows: jointly rank the sets {h*r}, {i*s}, {j*t}, {k*u}, h>=1, i>=1, j>=1, k>=1.
The position of n*r in the joint ranking is
n+[sn/r]+[tn/r]+[un/r], and likewise for the
positions of n*s, n*t, and n*u.

Crossrefs

Cf. A184912, A184917, A184918, A184919.

Programs

  • Mathematica
    r=1; s=2^(1/4); t=2^(1/2); u=2^(3/4);
a[n_]:=n+Floor[n*s/r]+Floor[n*t/r]+Floor[n*u/r];
b[n_]:=n+Floor[n*r/s]+Floor[n*t/s]+Floor[n*u/s];
c[n_]:=n+Floor[n*r/t]+Floor[n*s/t]+Floor[n*u/t];
d[n_]:=n+Floor[n*r/u]+Floor[n*s/u]+Floor[n*t/u];
Table[a[n],{n,1,120}]  (* A184916 *)
Table[b[n],{n,1,120}]  (* A184917 *)
Table[c[n],{n,1,120}]  (* A184918 *)
Table[d[n],{n,1,120}]  (* A184919 *)

Formula

a(n)=n+[sn/r]+[tn/r]+[un/r], where []=floor and
r=1, s=2^(1/4), t=s^2, u=s^3.

A184813 n+[nr/s]+[nt/s], where r=sqrt(2), s=sqrt(3), t=sqrt(5), and []=floor.

Original entry on oeis.org

2, 5, 8, 12, 15, 17, 21, 24, 27, 30, 33, 36, 39, 43, 46, 49, 51, 55, 58, 61, 65, 67, 70, 73, 77, 80, 83, 86, 89, 92, 96, 99, 101, 104, 108, 111, 114, 118, 120, 123, 126, 130, 133, 135, 139, 142, 145, 148, 152, 154, 157, 161, 164, 167, 170, 173, 176, 179, 183, 185, 188, 192, 195, 198
Offset: 1

Views

Author

Clark Kimberling, Jan 22 2011

Keywords

Comments

See A184812.

Links

Crossrefs

Cf. A184812, A184814.

Programs

  • Mathematica
    (See A184812.)
  • PARI
    a(n) = n + floor(n*sqrt(2/3)) + floor(n*sqrt(5/3)); \\ Michel Marcus, Jul 01 2017

A184814 n+[nr/t]+[ns/t], where r=sqrt(2), s=sqrt(3), t=sqrt(5), and []=floor.

Original entry on oeis.org

1, 4, 6, 9, 11, 13, 16, 19, 20, 23, 25, 28, 31, 32, 35, 38, 40, 42, 45, 47, 50, 52, 54, 57, 59, 62, 64, 66, 69, 71, 74, 76, 78, 81, 84, 85, 88, 91, 93, 95, 97, 100, 103, 105, 107, 110, 112, 115, 116, 119, 122, 124, 127, 129, 131, 134, 137, 138, 141, 143, 146, 149, 150, 153, 156, 158
Offset: 1

Views

Author

Clark Kimberling, Jan 22 2011

Keywords

Comments

See A184812.

Links

Crossrefs

Cf. A184812, A184813.

Programs

  • Mathematica
    (See A184812.)
  • PARI
    a(n) = n + floor(n*sqrt(2/5)) + floor(n*sqrt(3/5)); \\ Michel Marcus, Jul 01 2017

A184820 a(n) = n + floor(n/t) + floor(n/t^2), where t is the tribonacci constant.

Original entry on oeis.org

1, 3, 4, 7, 8, 10, 12, 14, 15, 17, 19, 21, 23, 25, 27, 28, 31, 32, 34, 35, 38, 39, 41, 44, 45, 47, 48, 51, 52, 54, 56, 58, 59, 62, 64, 65, 67, 69, 71, 72, 75, 76, 78, 80, 82, 84, 85, 88, 89, 91, 93, 95, 96, 98, 100, 102, 103, 106, 108, 109, 112, 113, 115, 116, 119, 120, 122, 124, 126, 128, 129, 132, 133, 135, 137, 139, 140, 143, 144, 146, 148, 150, 152, 153, 156, 157, 159, 161, 163, 164, 166, 169, 170, 172, 174, 176, 177, 179, 181, 183, 184, 187, 188, 190, 193, 194, 196, 197, 200, 201, 203, 205, 207, 208, 210, 213, 214, 216, 218, 220, 221, 224, 225, 227, 228, 231, 233, 234, 237, 238, 240, 242, 244, 245, 247
Offset: 1

Views

Author

Paul D. Hanna, Jan 22 2011

Keywords

Comments

This is one of three sequences that partition the positive integers.
Given t is the tribonacci constant, then the following sequences are disjoint:
. A184820(n) = n + [n/t] + [n/t^2],
. A184821(n) = n + [n*t] + [n/t],
. A184822(n) = n + [n*t] + [n*t^2], where []=floor.
This is a special case of Clark Kimberling's results given in A184812.

Examples

			Let t be the tribonacci constant, then t = 1 + 1/t + 1/t^2 where:
t = 1.8392867552..., t^2 = 3.3829757679..., t^3 = 6.2222625231...

Crossrefs

Cf. A184821, A184822, A184812, A058265; A184823, A184835.

Programs

  • PARI
    {a(n)=local(t=real(polroots(1+x+x^2-x^3)[1]));n+floor(n/t)+floor(n/t^2)}

Formula

Limit a(n)/n = t = 1.8392867552141611325518525646532866...
a(n) = n + floor(n*p/r) + floor(n*q/r), where p=t, q=t^2, r=t^3, and t is the tribonacci constant (see Clark Kimberling's formula in A184812).

A184823 a(n) = n + floor(n/t) + floor(n/t^2) + floor(n/t^3), where t is the tetranacci constant.

Original entry on oeis.org

1, 3, 4, 7, 8, 10, 11, 15, 16, 18, 19, 22, 23, 25, 28, 30, 31, 33, 35, 37, 38, 41, 43, 45, 46, 48, 51, 52, 55, 57, 59, 60, 62, 64, 66, 68, 70, 72, 74, 75, 78, 79, 82, 83, 86, 87, 89, 90, 93, 94, 97, 98, 101, 103, 104, 107, 108, 111, 112, 115, 116, 118, 119, 122, 124, 126, 128, 130, 131, 133, 135, 138, 139, 141, 143, 145, 146, 148, 151, 153, 155, 157, 159, 160, 162, 165, 167, 168, 170, 172, 174, 175, 178, 180, 182, 183, 186, 187, 189, 190, 194, 195, 197, 198, 201, 202, 204, 208, 209, 211, 212, 215, 216, 218, 220, 223, 224
Offset: 1

Views

Author

Paul D. Hanna, Jan 23 2011

Keywords

Comments

This is one of four sequences that partition the positive integers.
Given t is the tetranacci constant, then the following sequences are disjoint:
. A184823(n) = n + [n/t] + [n/t^2] + [n/t^3],
. A184824(n) = n + [n*t] + [n/t] + [n/t^2],
. A184825(n) = n + [n*t] + [n*t^2] + [n/t],
. A184826(n) = n + [n*t] + [n*t^2] + [n*t^3], where []=floor.
This is a special case of Clark Kimberling's results given in A184812.

Examples

			Let t be the tetranacci constant, then t = 1 + 1/t + 1/t^2 + 1/t^3 and:
t = 1.92756197548..., t^2 = 3.71549516932..., t^3 = 7.16184720848..., t^4 = 13.8049043532...

Crossrefs

Cf. A184824, A184825, A184826; A184820, A184835, A184812, A086088.

Programs

  • PARI
    {a(n)=local(t=real(polroots(1+x+x^2+x^3-x^4)[2])); n+floor(n/t)+floor(n/t^2)+floor(n/t^3)}

Formula

Limit a(n)/n = t = 1.9275619754829253042619058...
a(n) = n + floor(n*p/s) + floor(n*q/s) + floor(n*r/s), where p=t, q=t^2, r=t^3, s=t^4, and t is the tetranacci constant.

A184871 n+floor(ns/r)+floor(nt/r), where r=log(2), s=log(3), t=log(5).

Original entry on oeis.org

4, 9, 13, 19, 23, 28, 34, 38, 43, 48, 53, 58, 63, 68, 72, 78, 82, 87, 93, 97, 102, 107, 112, 117, 122, 127, 131, 137, 141, 146, 151, 156, 161, 165, 171, 176, 180, 186, 190, 195, 200, 205, 210, 215, 220, 224, 230, 235, 239, 245, 249, 254, 260, 264, 269, 274, 279, 283, 288, 294, 298, 303, 308, 313, 318, 323, 328, 332, 338, 342, 347, 353, 357, 362, 367, 372, 377, 382, 387, 391, 397, 401, 406, 412, 416, 421, 426, 431, 436, 440, 446, 450, 455, 460, 465, 470, 475, 480, 484, 490, 495, 499, 505, 509, 514, 520, 524, 529, 534, 539, 543, 549, 554, 558, 564, 568, 573, 578, 583, 588
Offset: 1

Views

Author

Clark Kimberling, Jan 23 2011

Keywords

Comments

This is one of three sequences that partition the positive integers. In general, suppose that r, s, t are positive real numbers for which the sets
{i/r: i>=1}, {j/s: j>=1}, {k/t: k>=1} are disjoint.
Let a(n) be the rank of n/r when all the numbers in the three sets are jointly ranked. Define b(n) and c(n) as the ranks of n/s and n/t. It is easy to prove that
a(n)=n+[ns/r]+[nt/r],
b(n)=n+[nr/s]+[nt/s],
c(n)=n+[nr/t]+[ns/t], where []=floor.
Taking r=log(2), s=log(3), t=log(5) yields
a=A184871, b=A184872, c=A184873.

Crossrefs

Cf. A184812, A184872, A184873, A184874 (primes in A184872).

Programs

  • Mathematica
    r=Log[2]; s=Log[3]; t=Log[5];
a[n_]:=n+Floor [n*s/r]+Floor[n*t/r];
b[n_]:=n+Floor [n*r/s]+Floor[n*t/s];
c[n_]:=n+Floor[n*r/t]+Floor[n*s/t];
Table[a[n],{n,1,120}]  (* A184871 *)
Table[b[n],{n,1,120}]  (* A184872 *)
Table[c[n],{n,1,120}]  (* A184873 *)

Extensions

Name corrected by Charles R Greathouse IV, Sep 04 2015
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