A184924 -id:A184924 - cp's OEIS frontend

A184925 n+[rn/s]+[tn/s]+[un/s], where []=floor and r=1, s=sqrt(3), t=sqrt(5), u=sqrt(7).

3, 8, 11, 17, 20, 25, 30, 34, 38, 42, 47, 51, 55, 61, 64, 69, 72, 78, 82, 86, 92, 95, 100, 103, 109, 113, 117, 122, 126, 130, 135, 139, 144, 147, 153, 156, 161, 166, 170, 175, 178, 184, 187, 192, 196, 201, 205, 209, 214, 218, 222, 228, 231, 236, 241, 245, 249, 253, 259, 262, 267, 271, 276, 279, 284, 289, 293, 297, 302, 306, 310, 314, 320, 324, 328, 333, 337, 342, 345, 351, 354, 359, 363, 368, 372, 377, 381, 385, 389, 394, 399, 403, 408, 412, 416, 420, 426, 429, 434, 438, 443, 446, 451, 456, 460, 464, 469, 473, 477, 483, 487, 491, 495, 500, 504, 508, 513, 518, 521, 526

Offset: 1

Clark Kimberling, Jan 26 2011

nonn

The sequences A184924-A184927 partition the positive integers:
  A184924: 6,14,21,28,37,44,52,59,...
  A184925: 3,8,11,17,20,25,30,34,...
  A184926: 2,5,9,12,15,19,23,26,29,...
  A184927: 1,4,7,10,13,16,18,22,24,...
Jointly rank the sets {h*r}, {i*s}, {j*t}, {k*u},
where h>=1, i>=1, j>=1, k>=1.  The position of n*s in the joint ranking is n+[rn/s]+[tn/s]+[un/s], and likewise for the positions of n*r, n*t, and n*u.

Cf. A184924, A184926, A184927.

r=1; s=3^(1/2); t=5^(1/2); u=7^(1/2);
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}]  (* A184924 *)
Table[b[n],{n,1,120}]  (* A184925 *)
Table[c[n],{n,1,120}]  (* A184926 *)
Table[d[n],{n,1,120}]  (* A184927 *)

A184926 n+[rn/t]+[sn/t]+[un/t], where []=floor and r=1, s=sqrt(3), t=sqrt(5), u=sqrt(7).

Original entry on oeis.org

2, 5, 9, 12, 15, 19, 23, 26, 29, 32, 36, 40, 43, 46, 49, 53, 57, 60, 63, 66, 70, 74, 77, 80, 84, 87, 90, 94, 97, 101, 104, 107, 111, 115, 118, 121, 124, 127, 132, 134, 138, 141, 145, 149, 152, 155, 159, 162, 164, 169, 172, 176, 179, 182, 186, 190, 193, 195, 199, 202, 207, 210, 213, 216, 220, 224, 226, 230, 233, 237, 240, 244, 247, 251, 254, 256, 261, 264, 268, 270, 274, 278, 282, 285, 288, 291, 294, 299, 301, 305, 308, 312, 316, 319, 322, 325, 329, 331, 336, 339, 343, 346, 349, 353, 356, 360, 362, 366, 369, 374, 376, 380, 383, 386, 391, 393, 397, 400, 404, 406

Offset: 1

Clark Kimberling, Jan 26 2011

nonn

The sequences A184924-A184927 partition the positive integers:
  A184924: 6,14,21,28,37,44,52,59,...
  A184925: 3,8,11,17,20,25,30,34,...
  A184926: 2,5,9,12,15,19,23,26,29,...
  A184927: 1,4,7,10,13,16,18,22,24,...
Jointly rank the sets {h*r}, {i*s}, {j*t}, {k*u},
where h>=1, i>=1, j>=1, k>=1.  The position of n*t in the joint ranking is n+[rn/t]+[sn/t]+[un/t], and likewise for the positions of n*r, n*s, and n*u.

Cf. A184924, A184925, A184927.

r=1; s=3^(1/2); t=5^(1/2); u=7^(1/2);
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}]  (* A184924 *)
Table[b[n],{n,1,120}]  (* A184925 *)
Table[c[n],{n,1,120}]  (* A184926 *)
Table[d[n],{n,1,120}]  (* A184927 *)

A184927 n+[rn/u]+[sn/u]+[tn/u], where []=floor and r=1, s=sqrt(3), t=sqrt(5), u=sqrt(7).

Original entry on oeis.org

1, 4, 7, 10, 13, 16, 18, 22, 24, 27, 31, 33, 35, 39, 41, 45, 48, 50, 54, 56, 58, 62, 65, 68, 71, 73, 76, 79, 81, 85, 88, 91, 93, 96, 99, 102, 105, 108, 110, 114, 116, 119, 123, 125, 129, 131, 133, 137, 140, 142, 146, 148, 151, 154, 157, 160, 163, 165, 168, 171, 174, 177, 180, 183, 185, 188, 191, 194, 198, 200, 203, 206, 208, 211, 215, 217, 221, 223, 225, 229, 232, 234, 238, 239, 243, 246, 248, 252, 255, 258, 260, 263, 266, 269, 272, 275, 277, 281, 283, 286, 290, 292, 295, 298, 300, 304, 307, 309, 313, 315, 317, 321, 323, 327, 330, 332, 335, 338, 340, 344

Offset: 1

Clark Kimberling, Jan 26 2011

The sequences A184924-A184927 partition the positive integers:
  A184924: 6,14,21,28,37,44,52,59,...
  A184925: 3,8,11,17,20,25,30,34,...
  A184926: 2,5,9,12,15,19,23,26,29,...
  A184927: 1,4,7,10,13,16,18,22,24,...
Jointly rank the sets {h*r}, {i*s}, {j*t}, {k*u},
where h>=1, i>=1, j>=1, k>=1.  The position of n*u in the joint ranking is n+[rn/u]+[sn/u]+[tn/u], and likewise for the positions of n*r, n*s, and n*t.

Cf. A184924, A184925, A184926.

r=1; s=3^(1/2); t=5^(1/2); u=7^(1/2);
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}]  (* A184924 *)
Table[b[n],{n,1,120}]  (* A184925 *)
Table[c[n],{n,1,120}]  (* A184926 *)
Table[d[n],{n,1,120}]  (* A184927 *)
With[{c=Sqrt[7]},Table[n+Floor[n/c]+Floor[(n Sqrt[3])/c]+Floor[(n Sqrt[5])/c],{n,120}]] (* Harvey P. Dale, Aug 28 2025 *)

Definition corrected by Georg Fischer, Jun 10 2020

A184928 a(n) = n + [sn/r] + [tn/r] + [un/r], where []=floor and r=sin(Pi/2), s=sin(Pi/3), t=sin(Pi/4), u=sin(Pi/5).

Original entry on oeis.org

1, 5, 8, 11, 14, 18, 21, 23, 27, 30, 33, 37, 40, 43, 45, 49, 52, 55, 59, 62, 65, 68, 71, 74, 77, 81, 84, 87, 91, 93, 96, 99, 103, 106, 109, 113, 116, 118, 121, 125, 128, 131, 135, 138, 140, 144, 147, 150, 153, 157, 160, 163, 166, 169, 172, 175, 179, 183, 185, 188, 191, 194, 198, 201, 204, 207, 211, 213, 216, 220, 223, 226, 229, 233, 236, 238, 242, 245, 248, 252, 255, 258, 260, 264, 267, 270, 274, 277, 280, 282, 286, 290, 292, 296, 299, 302, 306, 308, 312, 314, 318, 321, 324, 328, 330, 333, 336, 340, 344, 346, 350, 352, 355, 359, 362, 366, 368, 372, 375, 377

Offset: 1

Clark Kimberling, Jan 26 2011

nonn

The sequences A184924-A184928 partition the positive integers:
  A184928: 1, 5,  6, 11, 14, 18, 21, 23, 27, ...
  A184929: 2, 6, 10, 13, 17, 20, 24, 28, 32, ...
  A184930: 3, 7, 12, 16, 22, 25, 29, 34, 39, ...
  A184931: 4, 9, 15, 19, 26, 31, 36, 41, 47, ...
Jointly rank the sets {h*r}, {i*s}, {j*t}, {k*u}, where 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.

Cf. A184929, A184930, A184931; also, A184912, A184916, A184920, A184924.

r=Sin[Pi/2]; s=Sin[Pi/3]; t=Sin[Pi/4]; u=Sin[Pi/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}]  (* A184928 *)
Table[b[n],{n,1,120}]  (* A184929 *)
Table[c[n],{n,1,120}]  (* A184930 *)
Table[d[n],{n,1,120}]  (* A184931 *)

A184929 a(n) = n + [rn/s] + [tn/s] + [un/s], where []=floor and r=sin(Pi/2), s=sin(Pi/3), t=sin(Pi/4), u=sin(Pi/5).

Original entry on oeis.org

2, 6, 10, 13, 17, 20, 24, 28, 32, 35, 38, 42, 46, 50, 54, 57, 60, 64, 67, 72, 76, 78, 82, 86, 89, 94, 98, 101, 104, 108, 112, 115, 119, 123, 126, 130, 134, 137, 141, 145, 148, 152, 156, 158, 162, 167, 170, 174, 178, 180, 184, 189, 192, 196, 199, 203, 206, 210, 215, 217, 221, 225, 228, 232, 237, 239, 243, 247, 250, 254, 257, 261, 265, 269, 272, 276, 279, 283, 287, 291, 294, 297, 301, 305, 309, 313, 317, 319, 323, 327, 331, 335, 338, 341, 345, 349, 353, 357, 360, 363, 367, 371, 374, 378, 382, 385, 389, 393, 395, 400, 404, 408, 411, 415, 418, 421, 426, 430, 433, 436

Offset: 1

Clark Kimberling, Jan 26 2011

nonn

The sequences A184924-A184928 partition the positive integers:
  A184928: 1, 5,  6, 11, 14, 18, 21, 23, 27, ...
  A184929: 2, 6, 10, 13, 17, 20, 24, 28, 32, ...
  A184930: 3, 7, 12, 16, 22, 25, 29, 34, 39, ...
  A184931: 4, 9, 15, 19, 26, 31, 36, 41, 47, ...
Jointly rank the sets {h*r}, {i*s}, {j*t}, {k*u}, where h>=1, i>=1, j>=1, k>=1. The position of n*s in the joint ranking is n + [rn/s] + [tn/s] + [un/s], and likewise for the positions of n*r, n*t, and n*u.

Cf. A184928, A184930, A184931.

r=Sin[Pi/2]; s=Sin[Pi/3]; t=Sin[Pi/4]; u=Sin[Pi/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}]  (* A184928 *)
Table[b[n],{n,1,120}]  (* A184929 *)
Table[c[n],{n,1,120}]  (* A184930 *)
Table[d[n],{n,1,120}]  (* A184931 *)

A184930 a(n) = n + [rn/t] + [sn/t] + [un/t], where []=floor and r=sin(Pi/2), s=sin(Pi/3), t=sin(Pi/4), u=sin(Pi/5).

Original entry on oeis.org

3, 7, 12, 16, 22, 25, 29, 34, 39, 44, 48, 51, 56, 61, 66, 70, 75, 79, 83, 88, 92, 97, 102, 105, 110, 114, 120, 124, 129, 132, 136, 142, 146, 151, 155, 159, 164, 168, 173, 177, 182, 186, 190, 195, 200, 205, 209, 212, 218, 222, 227, 231, 235, 240, 244, 249, 253, 259, 263, 266, 271, 275, 281, 285, 289, 293, 298, 303, 307, 311, 316, 320, 325, 329, 334, 339, 343, 347, 351, 356, 361, 365, 369, 373, 379, 383, 388, 392, 396, 401, 405, 410, 414, 419, 423, 427, 432, 437, 442, 446, 449, 454, 459, 464, 468, 472, 477, 481, 486, 490, 494, 500, 503, 508, 512, 518, 522, 526, 530, 534

Offset: 1

Clark Kimberling, Jan 26 2011

nonn

The sequences A184924-A184928 partition the positive integers:
  A184928: 1, 5,  6, 11, 14, 18, 21, 23, 27, ...
  A184929: 2, 6, 10, 13, 17, 20, 24, 28, 32, ...
  A184930: 3, 7, 12, 16, 22, 25, 29, 34, 39, ...
  A184931: 4, 9, 15, 19, 26, 31, 36, 41, 47, ...
Jointly rank the sets {h*r}, {i*s}, {j*t}, {k*u}, where h>=1, i>=1, j>=1, k>=1. The position of n*t in the joint ranking is n + [rn/t] + [sn/t] + [un/t], and likewise for the positions of n*s, n*t, and n*u.

Cf. A184928, A184929, A184931.

r=Sin[Pi/2]; s=Sin[Pi/3]; t=Sin[Pi/4]; u=Sin[Pi/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}]  (* A184928 *)
Table[b[n],{n,1,120}]  (* A184929 *)
Table[c[n],{n,1,120}]  (* A184930 *)
Table[d[n],{n,1,120}]  (* A184931 *)

A184931 a(n) = n + [rn/u] + [sn/u] + [tn/u], where []=floor and r=sin(Pi/2), s=sin(Pi/3), t=sin(Pi/4), u=sin(Pi/5).

Original entry on oeis.org

4, 9, 15, 19, 26, 31, 36, 41, 47, 53, 58, 63, 69, 73, 80, 85, 90, 95, 100, 107, 111, 117, 122, 127, 133, 139, 143, 149, 154, 161, 165, 171, 176, 181, 187, 193, 197, 202, 208, 214, 219, 224, 230, 234, 241, 246, 251, 256, 262, 268, 273, 278, 284, 288, 295, 300, 304, 310, 315, 322, 326, 332, 337, 342, 348, 354, 358, 364, 370, 376, 380, 386, 391, 397, 402, 407, 413, 417, 424, 429, 434, 439, 445, 450, 456, 461, 467, 471, 478, 483, 488, 493, 499, 504, 509, 515, 520, 525, 531, 537, 541, 547, 552, 558, 563, 569, 574, 579, 585, 591, 595, 601, 606, 611, 617, 622, 628, 632, 639, 644

Offset: 1

Clark Kimberling, Jan 26 2011

nonn

The sequences A184924-A184928 partition the positive integers:
  A184928: 1, 5,  6, 11, 14, 18, 21, 23, 27, ...
  A184929: 2, 6, 10, 13, 17, 20, 24, 28, 32, ...
  A184930: 3, 7, 12, 16, 22, 25, 29, 34, 39, ...
  A184931: 4, 9, 15, 19, 26, 31, 36, 41, 47, ...
Jointly rank the sets {h*r}, {i*s}, {j*t}, {k*u}, where h>=1, i>=1, j>=1, k>=1. The position of n*u in the joint ranking is n + [rn/u] + [sn/u] + [tn/u], and likewise for the positions of n*r, n*s, and n*t.

Cf. A184928, A184929, A184930.

r=Sin[Pi/2]; s=Sin[Pi/3]; t=Sin[Pi/4]; u=Sin[Pi/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}]  (* A184928 *)
Table[b[n],{n,1,120}]  (* A184929 *)
Table[c[n],{n,1,120}]  (* A184930 *)
Table[d[n],{n,1,120}]  (* A184931 *)

cp's OEIS Frontend

A184925 n+[rn/s]+[tn/s]+[un/s], where []=floor and r=1, s=sqrt(3), t=sqrt(5), u=sqrt(7).

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A184926 n+[rn/t]+[sn/t]+[un/t], where []=floor and r=1, s=sqrt(3), t=sqrt(5), u=sqrt(7).

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A184927 n+[rn/u]+[sn/u]+[tn/u], where []=floor and r=1, s=sqrt(3), t=sqrt(5), u=sqrt(7).

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A184928 a(n) = n + [sn/r] + [tn/r] + [un/r], where []=floor and r=sin(Pi/2), s=sin(Pi/3), t=sin(Pi/4), u=sin(Pi/5).

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Mathematica

A184929 a(n) = n + [rn/s] + [tn/s] + [un/s], where []=floor and r=sin(Pi/2), s=sin(Pi/3), t=sin(Pi/4), u=sin(Pi/5).

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Mathematica

A184930 a(n) = n + [rn/t] + [sn/t] + [un/t], where []=floor and r=sin(Pi/2), s=sin(Pi/3), t=sin(Pi/4), u=sin(Pi/5).

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Mathematica

A184931 a(n) = n + [rn/u] + [sn/u] + [tn/u], where []=floor and r=sin(Pi/2), s=sin(Pi/3), t=sin(Pi/4), u=sin(Pi/5).

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