A351415 -id:A351415 - cp's OEIS frontend

A356104 a(n) = A000201(A022839(n)).

3, 6, 9, 12, 17, 21, 24, 27, 32, 35, 38, 42, 46, 50, 53, 56, 61, 64, 67, 71, 74, 79, 82, 85, 88, 93, 97, 100, 103, 108, 111, 114, 118, 122, 126, 129, 132, 135, 140, 144, 147, 150, 155, 158, 161, 165, 169, 173, 176, 179, 184, 187, 190, 194, 197, 202, 205, 208

Offset: 1

Clark Kimberling, Sep 08 2022

This is the first of four sequences that partition the positive integers. Suppose that u = (u(n)) and v = (v(n)) are increasing sequences of positive integers. Let u' and v' be their (increasing) complements, and consider these four sequences:
(1) u o v, defined by (u o v)(n) = u(v(n));
(2) u o v';
(3) u' o v;
(4) u' o v'.
Every positive integer is in exactly one of the four sequences. For the reverse composites, v o u, v' o u, v o u', v' o u', see A356217 to A356220.
Assume that if w is any of the sequences u, v, u', v', then lim_{n->oo} w(n)/n exists and defines the (limiting) density of w. For w = u,v,u',v', denote the densities by r,s,r',s'. Then the densities of sequences (1)-(4) exist, and
1/(r*r') + 1/(r*s') + 1/(s*s') + 1/(s*r') = 1.
For A356104, u, v, u', v', are the Beatty sequences given by u(n) = floor(n*(1+sqrt(5))/2) and v(n) = floor(n*sqrt(5)), so r = (1+sqrt(5))/2, s = sqrt(5), r' = (3+sqrt(5))/2, s' = (5 + sqrt(5))/4.

			(1)  u o v = (3, 6, 9, 12, 17, 21, 24, 27, 32, 35, 38, 42, 46, ...) = A356104
(2)  u o v' = (1, 4, 8, 11, 14, 16, 19, 22, 25, 29, 30, 33, 37, ...) = A356105
(3)  u' o v = (5, 10, 15, 20, 28, 34, 39, 44, 52, 57, 62, 68, ...) = A356106
(4)  u' o v' = (2, 7, 13, 18, 23, 26, 31, 36, 41, 47, 49, 54, ...) = A356107

Cf. u = A000201, u' = A001950, v = A022839, v' = A108598, A356105, A356106, A356107, A351415 (intersections), A356217 (reverse composites).

z = 1000;
u = Table[Floor[n*(1 + Sqrt[5])/2], {n, 1, z}];  (* A000201 *)
u1 = Complement[Range[Max[u]], u];  (* A001950 *)
v = Table[Floor[n*Sqrt[5]], {n, 1, z}];  (* A022839 *)
v1 = Complement[Range[Max[v]], v];  (* A108598 *)
zz = 120;
Table[u[[v[[n]]]], {n, 1, zz}]    (* A356104 *)
Table[u[[v1[[n]]]], {n, 1, zz}]   (* A356105 *)
Table[u1[[v[[n]]]], {n, 1, zz}]   (* A356106 *)
Table[u1[[v1[[n]]]], {n, 1, zz}]  (* A356107 *)

A356217 a(n) = A022839(A000201(n)).

Original entry on oeis.org

2, 6, 8, 13, 17, 20, 24, 26, 31, 35, 38, 42, 46, 49, 53, 55, 60, 64, 67, 71, 73, 78, 82, 84, 89, 93, 96, 100, 102, 107, 111, 114, 118, 122, 125, 129, 131, 136, 140, 143, 147, 149, 154, 158, 160, 165, 169, 172, 176, 178, 183, 187, 190, 194, 196, 201, 205, 207

Offset: 1

Clark Kimberling, Oct 02 2022

This is the first of four sequences that partition the positive integers. Suppose that u = (u(n)) and v = (v(n)) are increasing sequences of positive integers. Let u' and v' be their (increasing) complements, and consider these four sequences:
(1) v o u, defined by (v o u)(n) = v(u(n));
(2) v' o u;
(3) v o u';
(4) v' o u'.
Every positive integer is in exactly one of the four sequences. For the reverse composites, u o v, u o v', u' o v, u' o v', see A356104 to A356107.
Assume that if w is any of the sequences u, v, u', v', then lim_{n->oo} w(n)/n exists and defines the (limiting) density of w. For w = u,v,u',v', denote the densities by r,s,r',s'. Then the densities of sequences (1)-(4) exist, and
1/(r*r') + 1/(r*s') + 1/(s*s') + 1/(s*r') = 1.
For A356217 u, v, u', v', are the Beatty sequences given by u(n) = floor(n*(1+sqrt(5))/2) and v(n) = floor(n*sqrt(5)), so that r = (1+sqrt(5))/2, s = sqrt(5), r' = (3+sqrt(5))/2, s' = (5 + sqrt(5))/4.

			(1)  v o u = (2, 6, 8, 13, 17, 20, 24, 26, 31, 35, 38, 42, ...) = A356217
(2)  v' o u = (1, 5, 7, 10, 14, 16, 19, 21, 25, 28, 30, 34, ...) = A356218
(3)  v o u' = (4, 11, 15, 22, 29, 33, 40, 44, 51, 58, 62, 76, ...) = A190509
(4)  v' o u' = (3, 9, 12, 18, 23, 27, 32, 36, 41, 47, 50, 56, ...) = A356220

Cf. u = A000201, u' = A001950, v = A022839, v' = A108598, A351415 (intersections), A356104 (reverse composites), A356218, A190509, A356220.

z = 1000;
u = Table[Floor[n*(1 + Sqrt[5])/2], {n, 1, z}];  (* A000201 *)
u1 = Complement[Range[Max[u]], u];  (* A001950 *)
v = Table[Floor[n*Sqrt[5]], {n, 1, z}];  (* A022839 *)
v1 = Complement[Range[Max[v]], v];  (* A108598 *)
Table[v[[u[[n]]]], {n, 1, z/4}]   (* A356217 *)
Table[v1[[u[[n]]]], {n, 1, z/4}]  (* A356218 *)
Table[v[[u1[[n]]]], {n, 1, z/4}]  (* A190509 *)
Table[v1[[u1[[n]]]], {n, 1, z/4}] (* A356220 *)

from math import isqrt
def A356217(n): return isqrt(5*(n+isqrt(5*n**2)>>1)**2) # Chai Wah Wu, Oct 14 2022

A356107 a(n) = A001950(A108598(n)).

Original entry on oeis.org

2, 7, 13, 18, 23, 26, 31, 36, 41, 47, 49, 54, 60, 65, 70, 73, 78, 83, 89, 94, 96, 102, 107, 112, 117, 123, 125, 130, 136, 141, 146, 149, 154, 159, 164, 170, 172, 178, 183, 188, 193, 196, 201, 206, 212, 217, 222, 225, 230, 235, 240, 246, 248, 253, 259, 264

Offset: 1

Clark Kimberling, Oct 02 2022

This is the fourth of four sequences that partition the positive integers. See A356104.

			(1)  u o v = (3, 6, 9, 12, 17, 21, 24, 27, 32, 35, 38, 42, 46, ...) = A356104
(2)  u o v' = (1, 4, 8, 11, 14, 16, 19, 22, 25, 29, 30, 33, 37, ...) = A356105
(3)  u' o v = (5, 10, 15, 20, 28, 34, 39, 44, 52, 57, 62, 68, ...) = A356106
(4)  u' o v' = (2, 7, 13, 18, 23, 26, 31, 36, 41, 47, 49, 54, ...) = A356107

Cf. u = A000201, u' = A001950, v = A022839, v' = A108598, A356104, A356105, A356106, A351415 (intersections), A356217 (reverse composites).

z = 1000;
u = Table[Floor[n*(1 + Sqrt[5])/2], {n, 1, z}];  (* A000201 *)
u1 = Complement[Range[Max[u]], u];  (* A001950 *)
v = Table[Floor[n*Sqrt[5]], {n, 1, z}];  (* A022839 *)
v1 = Complement[Range[Max[v]], v];  (* A108598 *)
zz = 120;
Table[u[[v[[n]]]], {n, 1, zz}]    (* A356104 *)
Table[u[[v1[[n]]]], {n, 1, zz}]   (* A356105 *)
Table[u1[[v[[n]]]], {n, 1, zz}]   (* A356106 *)
Table[u1[[v1[[n]]]], {n, 1, zz}]  (* A356107 *)

A356220 a(n) = A108598(A001950(n)).

Original entry on oeis.org

3, 9, 12, 18, 23, 27, 32, 36, 41, 47, 50, 56, 61, 65, 70, 74, 79, 85, 88, 94, 97, 103, 108, 112, 117, 123, 126, 132, 135, 141, 146, 150, 155, 161, 164, 170, 173, 179, 184, 188, 193, 197, 202, 208, 211, 217, 222, 226, 231, 235, 240, 246, 249, 255, 258, 264

Offset: 1

Clark Kimberling, Nov 13 2022

This is the fourth of four sequences that partition the positive integers. See A356217.

			(1)  v o u = (2, 6, 8, 13, 17, 20, 24, 26, 31, 35, 38, 42, ...) = A356217
(2)  v' o u = (1, 5, 7, 10, 14, 16, 19, 21, 25, 28, 30, 34, ...) = A356218
(3)  v o u' = (4, 11, 15, 22, 29, 33, 40, 44, 51, 58, 62, 76, ...) = A190509
(4)  v' o u' = (3, 9, 12, 18, 23, 27, 32, 36, 41, 47, 50, 56, ...) = A356220

Cf. A000201, A001950, A022839, A108598, A351415 (intersections), A356104 (reverse composites), A356217, A356218, A356219.

z = 1000;
u = Table[Floor[n*(1 + Sqrt[5])/2], {n, 1, z}];  (* A000201 *)
u1 = Complement[Range[Max[u]], u];  (* A001950 *)
v = Table[Floor[n*Sqrt[5]], {n, 1, z}];  (* A022839 *)
v1 = Complement[Range[Max[v]], v];  (* A108598 *)
zz = 120;
Table[v[[u[[n]]]], {n, 1, z/4}]   (* A356217 *)
Table[v1[[u[[n]]]], {n, 1, z/4}]  (* A356218 *)
Table[v[[u1[[n]]]], {n, 1, z/4}]  (* A190509 *)
Table[v1[[u1[[n]]]], {n, 1, z/4}] (* A356220 *)

A356101 Intersection of A000201 and A022839.

Original entry on oeis.org

1, 3, 9, 12, 14, 16, 19, 21, 25, 27, 30, 32, 37, 43, 45, 48, 50, 56, 59, 61, 63, 66, 72, 74, 77, 79, 85, 88, 90, 92, 95, 97, 101, 103, 106, 108, 110, 113, 119, 121, 124, 126, 132, 135, 137, 139, 142, 144, 148, 150, 153, 155, 161, 166, 168, 171, 173, 177, 179

Offset: 1

Clark Kimberling, Sep 04 2022

This is the second of four sequences that partition the positive integers. See A351415.

			Starting with a general overview, suppose that u = (u(n)) and v = (v(n)) are increasing sequences of positive integers.  Let u' and v' be their complements, and assume that the following four sequences are infinite:
(1)  u ^ v = intersection of u and v (in increasing order);
(2)  u ^ v';
(3)  u' ^ v;
(4)  u' ^ v'.
Every positive integer is in exactly one of the four sequences. For A351415, u, v, u', v', are the Beatty sequences given by u(n) = floor(n*(1+sqrt(5))/2) and v(n) = floor(n*sqrt(5)), so r = (1+sqrt(5))/2, s = sqrt(5), r' = (3+sqrt(5))/2, s' = (5 + sqrt(5))/4.
(1)  u ^ v = (4, 6, 8, 11, 17, 22, 24, 29, 33, 35, 38, 40, 42, ...) =  A351415
(2)  u ^ v' = (1, 3, 9, 12, 14, 16, 19, 21, 25, 27, 30, 32, 37, ...) =  A356101
(3)  u' ^ v = (2, 13, 15, 20, 26, 31, 44, 49, 60, 62, 73, 78, ...) = A356102
(4)  u' ^ v' = (5, 7, 10, 18, 23, 28, 34, 36, 39, 41, 47, 52, 54, ...) = A356103

Cf. u = A000201, u' = A001950, v = A022839, v' = A108598, A351415, A356102, A356103, A356104 (results of composition instead of intersections), A190509 (composites, reversed order).

z = 200;
r = (1 + Sqrt[5])/2; u = Table[Floor[n*r], {n, 1, z}]  (* A000201 *)
u1 = Take[Complement[Range[1000], u], z]  (* A001950 *)
r1 = Sqrt[5]; v = Table[Floor[n*r1], {n, 1, z}]  (* A022839 *)
v1 = Take[Complement[Range[1000], v], z]  (* A108598 *)
Intersection[u, v]   (* A351415 *)
Intersection[u, v1]  (* A356101 *)
Intersection[u1, v]  (* A356102 *)
Intersection[u1, v1] (* A356103 *)

A356102 Intersection of A001950 and A022839.

Original entry on oeis.org

2, 13, 15, 20, 26, 31, 44, 49, 60, 62, 73, 78, 89, 91, 96, 102, 107, 109, 120, 125, 136, 138, 143, 149, 154, 167, 172, 178, 183, 185, 196, 201, 212, 214, 219, 225, 230, 243, 248, 259, 261, 272, 277, 290, 295, 301, 306, 308, 319, 324, 326, 328, 330, 333, 335

Offset: 1

Clark Kimberling, Sep 04 2022

This is the third of four sequences that partition the positive integers. See A351415.

			Starting with a general overview, suppose that u = (u(n)) and v = (v(n)) are increasing sequences of positive integers.  Let u' and v' be their complements, and assume that the following four sequences are infinite:
(1)  u ^ v = intersection of u and v (in increasing order);
(2)  u ^ v';
(3)  u' ^ v;
(4)  u' ^ v'.
Every positive integer is in exactly one of the four sequences. For A351415, u, v, u', v', are the Beatty sequences given by u(n) = floor(n*(1+sqrt(5))/2) and v(n) = floor(n*sqrt(5)), so that r = (1+sqrt(5))/2, s = sqrt(5), r' = (3+sqrt(5))/2, s' = (5 + sqrt(5))/4.
(1)  u ^ v = (4, 6, 8, 11, 17, 22, 24, 29, 33, 35, 38, 40, 42, ...) =  A351415
(2)  u ^ v' = (1, 3, 9, 12, 14, 16, 19, 21, 25, 27, 30, 32, 37, ...) =  A356101
(3)  u' ^ v = (2, 13, 15, 20, 26, 31, 44, 49, 60, 62, 73, 78, ...) = A356102
(4)  u' ^ v' = (5, 7, 10, 18, 23, 28, 34, 36, 39, 41, 47, 52, 54, ...) = A356103

Cf. u = A000201, u' = A001950, v = A022839, v' = A108598, A351415, A356101, A356103, A356104 (results of composition instead of intersections), A190509 (composites, reversed order).

z = 200;
r = (1 + Sqrt[5])/2; u = Table[Floor[n*r], {n, 1, z}]  (* A000201 *)
u1 = Take[Complement[Range[1000], u], z]  (* A001950 *)
r1 = Sqrt[5]; v = Table[Floor[n*r1], {n, 1, z}]  (* A022839 *)
v1 = Take[Complement[Range[1000], v], z]  (* A108598 *)
Intersection[u, v]   (* A351415 *)
Intersection[u, v1]  (* A356101 *)
Intersection[u1, v]  (* A356102 *)
Intersection[u1, v1] (* A356103 *)

A356103 Intersection of A001950 and A108598.

Original entry on oeis.org

5, 7, 10, 18, 23, 28, 34, 36, 39, 41, 47, 52, 54, 57, 65, 68, 70, 75, 81, 83, 86, 94, 99, 104, 112, 115, 117, 123, 128, 130, 133, 141, 146, 151, 157, 159, 162, 164, 170, 175, 180, 188, 191, 193, 198, 204, 206, 209, 217, 222, 227, 233, 235, 238, 240, 246, 251

Offset: 1

Clark Kimberling, Sep 04 2022

This is the fourth of four sequences that partition the positive integers. See A351415.

			Starting with a general overview, suppose that u = (u(n)) and v = (v(n)) are increasing sequences of positive integers.  Let u' and v' be their complements, and assume that the following four sequences are infinite:
(1)  u ^ v = intersection of u and v (in increasing order);
(2)  u ^ v';
(3)  u' ^ v;
(4)  u' ^ v'.
Every positive integer is in exactly one of the four sequences. For A351415, u, v, u', v', are the Beatty sequences given by u(n) = floor(n*(1+sqrt(5))/2) and v(n) = floor(n*sqrt(5)), so that r = (1+sqrt(5))/2, s = sqrt(5), r' = (3+sqrt(5))/2, s' = (5 + sqrt(5))/4.
(1)  u ^ v = (4, 6, 8, 11, 17, 22, 24, 29, 33, 35, 38, 40, 42, ...) =  A351415
(2)  u ^ v' = (1, 3, 9, 12, 14, 16, 19, 21, 25, 27, 30, 32, 37, ...) =  A356101
(3)  u' ^ v = (2, 13, 15, 20, 26, 31, 44, 49, 60, 62, 73, 78, ...) = A356102
(4)  u' ^ v' = (5, 7, 10, 18, 23, 28, 34, 36, 39, 41, 47, 52, 54, ...) = A356103

Cf. u = A000201, u' = A001950, v = A022839, v' = A108598, A351415, A356101, A356102, A356104 (results of composition instead of intersections), A190509 (composites, reversed order).

z = 200;
r = (1 + Sqrt[5])/2; u = Table[Floor[n*r], {n, 1, z}]  (* A000201 *)
u1 = Take[Complement[Range[1000], u], z]  (* A001950 *)
r1 = Sqrt[5]; v = Table[Floor[n*r1], {n, 1, z}]  (* A022839 *)
v1 = Take[Complement[Range[1000], v], z]  (* A108598 *)
Intersection[u, v]   (* A351415 *)
Intersection[u, v1]  (* A356101 *)
Intersection[u1, v]  (* A356102 *)
Intersection[u1, v1] (* A356103 *)

A356105 a(n) = A000201(A108598(n)).

Original entry on oeis.org

1, 4, 8, 11, 14, 16, 19, 22, 25, 29, 30, 33, 37, 40, 43, 45, 48, 51, 55, 58, 59, 63, 66, 69, 72, 76, 77, 80, 84, 87, 90, 92, 95, 98, 101, 105, 106, 110, 113, 116, 119, 121, 124, 127, 131, 134, 137, 139, 142, 145, 148, 152, 153, 156, 160, 163, 166, 168, 171

Offset: 1

Clark Kimberling, Sep 08 2022

This is the second of four sequences that partition the positive integers. See A356104.

			(1)  u o v = (3, 6, 9, 12, 17, 21, 24, 27, 32, 35, 38, 42, 46, ...) = A356104
(2)  u o v' = (1, 4, 8, 11, 14, 16, 19, 22, 25, 29, 30, 33, 37, ...) = A356105
(3)  u' o v = (5, 10, 15, 20, 28, 34, 39, 44, 52, 57, 62, 68, ...) = A356106
(4)  u' o v' = (2, 7, 13, 18, 23, 26, 31, 36, 41, 47, 49, 54, ...) = A356107

Cf. u = A000201, u' = A001950, v = A022839, v' = A108598, A356104, A356106, A356107, A351415 (intersections), A356217 (reverse composites).

z = 1000;
u = Table[Floor[n*(1 + Sqrt[5])/2], {n, 1, z}];  (* A000201 *)
u1 = Complement[Range[Max[u]], u];  (* A001950 *)
v = Table[Floor[n*Sqrt[5]], {n, 1, z}];  (* A022839 *)
v1 = Complement[Range[Max[v]], v];  (* A108598 *)
zz = 120;
Table[u[[v[[n]]]], {n, 1, zz}]    (* A356104 *)
Table[u[[v1[[n]]]], {n, 1, zz}]   (* A356105 *)
Table[u1[[v[[n]]]], {n, 1, zz}]   (* A356106 *)
Table[u1[[v1[[n]]]], {n, 1, zz}]  (* A356107 *)

A356218 a(n) = A108598(A000201(n)).

Original entry on oeis.org

1, 5, 7, 10, 14, 16, 19, 21, 25, 28, 30, 34, 37, 39, 43, 45, 48, 52, 54, 57, 59, 63, 66, 68, 72, 75, 77, 81, 83, 86, 90, 92, 95, 99, 101, 104, 106, 110, 113, 115, 119, 121, 124, 128, 130, 133, 137, 139, 142, 144, 148, 151, 153, 157, 159, 162, 166, 168, 171

Offset: 1

Clark Kimberling, Oct 02 2022

This is the second of four sequences that partition the positive integers. See A356217.

			(1)  v o u = (2, 6, 8, 13, 17, 20, 24, 26, 31, 35, 38, 42, ...) = A356217
(2)  v' o u = (1, 5, 7, 10, 14, 16, 19, 21, 25, 28, 30, 34, ...) = A356218
(3)  v o u' = (4, 11, 15, 22, 29, 33, 40, 44, 51, 58, 62, 76, ...) = A190509
(4)  v' o u' = (3, 9, 12, 18, 23, 27, 32, 36, 41, 47, 50, 56, ...) = A356220

Cf. u = A000201, u' = A001950, v = A022839, v' = A108598, A351415 (intersections), A356104 (reverse composites), A356217, A190509, A356220.

z = 1000;
u = Table[Floor[n*(1 + Sqrt[5])/2], {n, 1, z}];  (* A000201 *)
u1 = Complement[Range[Max[u]], u];  (* A001950 *)
v = Table[Floor[n*Sqrt[5]], {n, 1, z}];  (* A022839 *)
v1 = Complement[Range[Max[v]], v];  (* A108598 *)
zz = 120;
Table[v[[u[[n]]]], {n, 1, z/4}]   (* A356217 *)
Table[v1[[u[[n]]]], {n, 1, z/4}]  (* A356218 *)
Table[v[[u1[[n]]]], {n, 1, z/4}]  (* A190509 *)
Table[v1[[u1[[n]]]], {n, 1, z/4}] (* A356220 *)

A356106 a(n) = A001950(A022839(n)).

Original entry on oeis.org

5, 10, 15, 20, 28, 34, 39, 44, 52, 57, 62, 68, 75, 81, 86, 91, 99, 104, 109, 115, 120, 128, 133, 138, 143, 151, 157, 162, 167, 175, 180, 185, 191, 198, 204, 209, 214, 219, 227, 233, 238, 243, 251, 256, 261, 267, 274, 280, 285, 290, 298, 303, 308, 314, 319

Offset: 1

Clark Kimberling, Sep 08 2022

This is the third of four sequences that partition the positive integers. See A356104.

			(1)  u o v = (3, 6, 9, 12, 17, 21, 24, 27, 32, 35, 38, 42, 46, ...) = A356104
(2)  u o v' = (1, 4, 8, 11, 14, 16, 19, 22, 25, 29, 30, 33, 37, ...) = A356105
(3)  u' o v = (5, 10, 15, 20, 28, 34, 39, 44, 52, 57, 62, 68, ...) = this sequence
(4)  u' o v' = (2, 7, 13, 18, 23, 26, 31, 36, 41, 47, 49, 54, ...) = A356107

Cf. u = A000201, u' = A001950, v = A022839, v' = A108598, A356104, A356105, A356107, A351415 (intersections), A356217 (reverse composites).

z = 1000;
u = Table[Floor[n*(1 + Sqrt[5])/2], {n, 1, z}];  (* A000201 *)
u1 = Complement[Range[Max[u]], u];  (* A001950 *)
v = Table[Floor[n*Sqrt[5]], {n, 1, z}];  (* A022839 *)
v1 = Complement[Range[Max[v]], v];  (* A108598 *)
zz = 120;
Table[u[[v[[n]]]], {n, 1, zz}]    (* A356104 *)
Table[u[[v1[[n]]]], {n, 1, zz}]   (* A356105 *)
Table[u1[[v[[n]]]], {n, 1, zz}]   (* this sequence *)
Table[u1[[v1[[n]]]], {n, 1, zz}]  (* A356107 *)

Definition corrected by Georg Fischer, May 24 2024

cp's OEIS Frontend

A356104 a(n) = A000201(A022839(n)).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A356217 a(n) = A022839(A000201(n)).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Python

A356107 a(n) = A001950(A108598(n)).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A356220 a(n) = A108598(A001950(n)).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A356101 Intersection of A000201 and A022839.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A356102 Intersection of A001950 and A022839.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A356103 Intersection of A001950 and A108598.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A356105 a(n) = A000201(A108598(n)).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A356218 a(n) = A108598(A000201(n)).

Views

Author

Keywords

Comments

Examples