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.

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A378185 a(n) = n + floor(n*r/s) + floor(n*r/t), where r=2^(1/4), s=2^(1/2), t=2^(3/4).

Original entry on oeis.org

2, 5, 8, 11, 14, 18, 20, 23, 26, 29, 33, 36, 38, 41, 44, 48, 51, 54, 56, 59, 62, 66, 69, 72, 75, 77, 81, 84, 87, 90, 93, 96, 99, 102, 105, 108, 112, 114, 117, 120, 123, 126, 130, 132, 135, 138, 141, 145, 148, 151, 153, 156, 160, 163, 166, 169, 171, 174, 178
Offset: 1

Views

Author

Clark Kimberling, Jan 13 2025

Keywords

Comments

The sequences A378142, A328185, A379510, partition the positive integers (A000027), as proved at A184812:
A378142: 3,6,10,13,17,21,24,28,32,35,...
A328185: 2,5,8,11,14,18,20,23,26,29,,...
A379510: 1,4,7,9,12,15,16,19,22,25,27,...
For each k in A000027, write "a" if k=A378142(n) for some n, "b" if k=A328185(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, A184812, A378142, 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] + [n/w], where w = 2^(1/4) and [ ] = floor.

A379510 a(n) = n + floor(n*r/t) + floor(n*s/t), where r=2^(1/4), s=2^(1/2), t=2^(3/4).

Original entry on oeis.org

1, 4, 7, 9, 12, 15, 16, 19, 22, 25, 27, 30, 32, 34, 37, 40, 43, 45, 47, 50, 52, 55, 58, 60, 63, 65, 68, 70, 73, 76, 78, 80, 83, 86, 88, 91, 94, 95, 98, 101, 103, 106, 109, 111, 113, 116, 119, 121, 124, 127, 129, 131, 134, 137, 139, 142, 144, 147, 149, 152
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 k in A000027, 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, A184812, A378142, A378185.

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

A184836 a(n) = n + floor(n*t) + floor(n/t) + floor(n/t^2) + floor(n/t^3), where t is the pentanacci constant.

Original entry on oeis.org

2, 6, 9, 14, 17, 21, 24, 30, 33, 37, 40, 45, 48, 52, 55, 61, 64, 68, 71, 76, 79, 83, 87, 92, 95, 99, 102, 107, 110, 113, 118, 122, 125, 129, 133, 137, 140, 145, 149, 153, 156, 160, 164, 168, 171, 176, 180, 184, 187, 191, 195, 199, 202, 207, 211, 215, 218, 223, 226, 229, 234, 238, 242, 245, 249, 253, 257, 260, 265, 269, 273, 276, 280, 284, 288, 292, 296, 300, 304, 307, 311, 315, 319, 323, 327, 331, 335, 338, 342, 345, 349, 353, 358, 361, 365, 368, 373, 376, 381, 384, 389, 392, 396, 399, 404, 407, 412, 415, 420, 423
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^2 = 1 + t + 1/t + 1/t^2 + 1/t^3,
t = 1.965948236645..., t^2 = 3.864952469169..., t^3 = 7.598296491482..., t^4 = 14.93785758893..., t^5 = 29.36705478623...

Crossrefs

Cf. A184835, A184837, A184838, A184839; A184812, A103814.

Programs

  • 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)+floor(n/t^2)+floor(n/t^3)}

Formula

Limit a(n)/n = t^2 = 3.8649524691694932164414964...
a(n) = n + floor(n*p/s) + floor(n*q/s) + floor(n*r/s) + floor(n*u/s), where p=t, q=t^2, r=t^3, s=t^4, u=t^5, and t is the pentanacci constant.

A184837 a(n) = n + floor(n*t) + floor(n*t^2) + floor(n/t) + floor(n/t^2), where t is the pentanacci constant.

Original entry on oeis.org

5, 13, 20, 29, 36, 44, 51, 59, 66, 74, 81, 90, 97, 105, 111, 120, 127, 135, 142, 151, 158, 166, 172, 181, 188, 196, 203, 212, 219, 225, 233, 241, 248, 256, 264, 272, 279, 286, 294, 302, 309, 317, 325, 333, 339, 347, 355, 363, 370, 378, 386, 393, 400, 408, 416, 424, 431, 440, 447, 453, 461, 469, 477, 484, 492, 500, 507, 514, 522, 530, 538, 545, 553, 561, 568, 575, 583, 591, 599, 606, 614, 621, 629, 636, 644, 652, 660, 667, 674, 681, 689, 696, 705, 712, 720, 727, 735, 742, 750, 757, 766, 773, 781, 787, 796, 803
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^3 = 1 + t + t^2 + 1/t + 1/t^2,
t = 1.965948236645..., t^2 = 3.864952469169..., t^3 = 7.598296491482..., t^4 = 14.93785758893..., t^5 = 29.36705478623...

Crossrefs

Cf. A184835, A184836, A184838, A184839; A184812, A103814.

Programs

  • 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)+floor(n/t^2)}

Formula

Limit a(n)/n = t^3 = 7.5982964914823797216620775...
a(n) = n + floor(n*p/r) + floor(n*q/r) + floor(n*s/r) + floor(n*u/r), where p=t, q=t^2, r=t^3, s=t^4, u=t^5, and t is the pentanacci constant.

A184838 a(n) = n + floor(n*t) + floor(n*t^2) + floor(n*t^3) + floor(n/t), where t is the pentanacci constant.

Original entry on oeis.org

12, 28, 42, 58, 72, 88, 103, 117, 132, 147, 162, 178, 192, 208, 221, 237, 252, 267, 282, 297, 312, 328, 341, 357, 371, 387, 402, 417, 432, 445, 460, 476, 490, 506, 520, 536, 551, 565, 580, 595, 610, 626, 640, 656, 669, 685, 700, 715, 730, 745, 760, 775, 789, 805, 819, 835, 850, 865, 880, 893, 909, 924, 939, 954, 969, 984, 999, 1013, 1029, 1043, 1059, 1074, 1089, 1104, 1118, 1133, 1149, 1163, 1179, 1193, 1209, 1223, 1238, 1253, 1268, 1283, 1299, 1313, 1327, 1341, 1357, 1372, 1387, 1402, 1417, 1432, 1447
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^4 = 1 + t + t^2 + t^3 + 1/t,
t = 1.965948236645..., t^2 = 3.864952469169..., t^3 = 7.598296491482..., t^4 = 14.93785758893..., t^5 = 29.36705478623...

Crossrefs

Cf. A184835, A184836, A184837, A184839; A184812, A103814.

Programs

  • 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)}

Formula

Limit a(n)/n = t^4 = 14.937857588939362411757354...
a(n) = n + floor(n*p/q) + floor(n*r/q) + floor(n*s/q) + floor(n*u/q), where p=t, q=t^2, r=t^3, s=t^4, u=t^5, and t is the pentanacci constant.

A184839 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

26, 56, 85, 115, 144, 174, 204, 232, 262, 291, 321, 351, 380, 410, 438, 468, 497, 526, 556, 585, 615, 645, 673, 703, 732, 762, 792, 821, 851, 878, 908, 938, 966, 996, 1025, 1055, 1085, 1113, 1143, 1172, 1202, 1232, 1261, 1291, 1319, 1349, 1379, 1408, 1437, 1466, 1496, 1525, 1554, 1584, 1613, 1643, 1673, 1702, 1731, 1759, 1789, 1819, 1848, 1878, 1906, 1936, 1965, 1994, 2024, 2053, 2083, 2113, 2142, 2172, 2200, 2230, 2260, 2289, 2319, 2348, 2377, 2406, 2435, 2465, 2494, 2524, 2554, 2583, 2611, 2640, 2670
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^5 = 1 + t + t^2 + t^3 + t^4,
t = 1.965948236645..., t^2 = 3.864952469169..., t^3 = 7.598296491482..., t^4 = 14.93785758893..., t^5 = 29.36705478623...

Links

Crossrefs

Cf. A184835, A184836, A184837, A184838; A184812, A103814.

Programs

  • Mathematica
    With[{t=Root[x^5-x^4-x^3-x^2-x-1,1]},Table[n+Total@@Through[ Floor[ n*t^Range[4]]],{n,100}]] (* Harvey P. Dale, Dec 12 2019 *)
  • 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^5 = 29.367054786236720687050865...
a(n) = n + floor(n*q/p) + floor(n*r/p) + floor(n*s/p) + floor(n*u/p), where p=t, q=t^2, r=t^3, s=t^4, u=t^5, and t is the pentanacci constant.

A184915 n+[rn/u]+[sn/u]+[tn/u], where []=floor and r=2^(1/5), s=r^2, t=r^3, u=r^4.

Original entry on oeis.org

1, 5, 8, 12, 15, 18, 22, 25, 27, 31, 35, 38, 41, 45, 48, 51, 54, 57, 61, 65, 67, 71, 75, 77, 80, 84, 87, 91, 94, 97, 100, 104, 107, 110, 114, 117, 121, 124, 126, 130, 134, 136, 140, 144, 147, 150, 153, 156, 160, 162, 166, 170, 173, 176, 179, 182, 186, 189, 192, 196, 200, 201, 205, 209, 212, 216, 219, 222, 226, 229, 231, 235, 239, 242, 245, 249, 252, 255, 258, 261, 265, 269, 271, 275, 278, 281, 284, 288, 291, 295, 298, 301, 304, 308, 310, 314, 317, 321, 325, 327, 330, 334, 337, 340, 344, 347, 351, 354, 356, 360, 364, 366, 370, 374, 377, 379, 383, 386, 390, 393
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*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.

Crossrefs

Cf. A184812, A184912, A184913, A184914.

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 *)

A184815 Numbers m such that prime(m) is of the form k+floor(ks/r)+floor(kt/r), where r=sqrt(2), s=sqrt(3), t=sqrt(5).

Original entry on oeis.org

2, 4, 10, 12, 13, 16, 22, 29, 30, 36, 42, 44, 45, 49, 52, 57, 59, 60, 64, 70, 71, 76, 82, 84, 90, 91, 92, 97, 101, 103, 108, 111, 114, 115, 119, 123, 125, 138, 140, 142, 149, 150, 165, 171, 178, 180, 182, 185, 189, 191, 192, 193, 195, 198, 205, 211, 215, 217, 220, 222, 224, 233, 235, 236, 247, 248, 249, 252, 254, 255, 264, 265, 269, 273, 286, 295, 301, 302, 306, 307, 309, 316, 318, 325, 326, 327, 328, 329, 332, 336
Offset: 1

Views

Author

Clark Kimberling, Jan 23 2011

Keywords

Comments

A184815, A184816, and A184817 partition the primes:
A184815: 3,7,29,37,... of the form n+[ns/r]+[nt/r].
A184816: 2,5,17,... of the form n+[nr/s]+[nt/s].
A184817: 11,13,19,23,31,... of the form n+[nr/t]+[ns/t].
The Mathematica code can be easily modified to print primes in the three classes.

Examples

			See A184812.

Links

Crossrefs

Cf. A184812, A184816, A184817.

Programs

  • Mathematica
    r=2^(1/2); s=3^(1/2); t=5^(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}]  (* A184812 *)
Table[b[n],{n,1,120}]  (* A184813 *)
Table[c[n],{n,1,120}]  (* A184814 *)
t1={};Do[If[PrimeQ[a[n]], AppendTo[t1,a[n]]],{n,1,600}];t1;
t2={};Do[If[PrimeQ[a[n]], AppendTo[t2,n]],{n,1,600}];t2;
t3={};Do[If[MemberQ[t1,Prime[n]],AppendTo[t3,n]],{n,1,600}];t3
t4={};Do[If[PrimeQ[b[n]], AppendTo[t4,b[n]]],{n,1,600}];t4;
t5={};Do[If[PrimeQ[b[n]], AppendTo[t5,n]],{n,1,600}];t5;
t6={};Do[If[MemberQ[t4,Prime[n]],AppendTo[t6,n]],{n,1,600}];t6
t7={};Do[If[PrimeQ[c[n]], AppendTo[t7,c[n]]],{n,1,600}];t7;
t8={};Do[If[PrimeQ[c[n]], AppendTo[t8,n]],{n,1,600}];t8;
t9={};Do[If[MemberQ[t7,Prime[n]],AppendTo[t9,n]],{n,1,600}];t9
(* Lists t3, t6, t9 match A184815, A184816, A184817. *)

A184816 Numbers m such that prime(m) is of the form k+floor(kr/s)+floor(kt/s), where r=sqrt(2), s=sqrt(3), t=sqrt(5).

Original entry on oeis.org

1, 3, 7, 14, 18, 19, 21, 23, 24, 26, 34, 37, 39, 40, 41, 50, 53, 54, 55, 56, 65, 68, 69, 72, 78, 80, 81, 83, 86, 93, 95, 96, 98, 105, 106, 109, 113, 117, 124, 126, 129, 131, 133, 135, 137, 139, 143, 145, 148, 152, 157, 158, 159, 160, 161, 162, 168, 169, 172, 173, 174, 176, 183, 187, 190, 197, 200, 207, 208, 212, 214, 219, 229, 232, 234, 238, 242, 243, 245, 246, 257, 258, 259, 266, 267, 268, 270, 275, 276, 278, 279, 280, 281, 284
Offset: 1

Views

Author

Clark Kimberling, Jan 23 2011

Keywords

Comments

See A184812 and A184815.

Links

Crossrefs

Cf. A184812, A184815, A184817.

Programs

  • Mathematica
    r=2^(1/2); s=3^(1/2); t=5^(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}]  (* A184812 *)
Table[b[n],{n,1,120}]  (* A184813 *)
Table[c[n],{n,1,120}]  (* A184814 *)
t1={};Do[If[PrimeQ[a[n]], AppendTo[t1,a[n]]],{n,1,600}];t1;
t2={};Do[If[PrimeQ[a[n]], AppendTo[t2,n]],{n,1,600}];t2;
t3={};Do[If[MemberQ[t1,Prime[n]],AppendTo[t3,n]],{n,1,600}];t3
t4={};Do[If[PrimeQ[b[n]], AppendTo[t4,b[n]]],{n,1,600}];t4;
t5={};Do[If[PrimeQ[b[n]], AppendTo[t5,n]],{n,1,600}];t5;
t6={};Do[If[MemberQ[t4,Prime[n]],AppendTo[t6,n]],{n,1,600}];t6
t7={};Do[If[PrimeQ[c[n]], AppendTo[t7,c[n]]],{n,1,600}];t7;
t8={};Do[If[PrimeQ[c[n]], AppendTo[t8,n]],{n,1,600}];t8;
t9={};Do[If[MemberQ[t7,Prime[n]],AppendTo[t9,n]],{n,1,600}];t9
(* Lists t3, t6, t9 match A184815, A184816, A184817. *)
PrimePi/@Select[Table[k+Floor[(k Sqrt[2])/Sqrt[3]]+Floor[(k Sqrt[5])/Sqrt[3]],{k,600}],PrimeQ] (* Harvey P. Dale, Apr 25 2023 *)

A184817 Numbers m such that prime(m) is of the form k+floor(kr/t)+floor(ks/t), where r=sqrt(2), s=sqrt(3), t=sqrt(5).

Original entry on oeis.org

5, 6, 8, 9, 11, 15, 17, 20, 25, 27, 28, 31, 32, 33, 35, 38, 43, 46, 47, 48, 51, 58, 61, 62, 63, 66, 67, 73, 74, 75, 77, 79, 85, 87, 88, 89, 94, 99, 100, 102, 104, 107, 110, 112, 116, 118, 120, 121, 122, 127, 128, 130, 132, 134, 136, 141, 144, 146, 147, 151, 153, 154, 155, 156, 163, 164, 166, 167, 170, 175, 177, 179, 181, 184, 186, 188, 194, 196, 199, 201, 202, 203, 204, 206, 209, 210, 213, 216, 218, 221, 223, 225, 226, 227, 228
Offset: 1

Views

Author

Clark Kimberling, Jan 23 2011

Keywords

Comments

See A184812 and A184814.

Links

Crossrefs

Cf. A184812, A184815, A184816, A184817.

Programs

  • Mathematica
    r=2^(1/2); s=3^(1/2); t=5^(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}]  (* A184812 *)
Table[b[n],{n,1,120}]  (* A184813 *)
Table[c[n],{n,1,120}]  (* A184814 *)
t1={};Do[If[PrimeQ[a[n]], AppendTo[t1,a[n]]],{n,1,600}];t1;
t2={};Do[If[PrimeQ[a[n]], AppendTo[t2,n]],{n,1,600}];t2;
t3={};Do[If[MemberQ[t1,Prime[n]],AppendTo[t3,n]],{n,1,600}];t3
t4={};Do[If[PrimeQ[b[n]], AppendTo[t4,b[n]]],{n,1,600}];t4;
t5={};Do[If[PrimeQ[b[n]], AppendTo[t5,n]],{n,1,600}];t5;
t6={};Do[If[MemberQ[t4,Prime[n]],AppendTo[t6,n]],{n,1,600}];t6
t7={};Do[If[PrimeQ[c[n]], AppendTo[t7,c[n]]],{n,1,600}];t7;
t8={};Do[If[PrimeQ[c[n]], AppendTo[t8,n]],{n,1,600}];t8;
t9={};Do[If[MemberQ[t7,Prime[n]],AppendTo[t9,n]],{n,1,600}];t9
(* Lists t3, t6, t9 match A184815, A184816, A184817. *)
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