A026530 -id:A026530 - cp's OEIS frontend

A360405 a(n) = A360393(A356133(n)).

2, 6, 15, 27, 34, 45, 55, 60, 69, 81, 91, 96, 108, 114, 124, 135, 142, 153, 163, 168, 180, 186, 195, 208, 217, 222, 231, 244, 249, 262, 271, 276, 285, 297, 307, 312, 324, 330, 339, 352, 361, 366, 375, 387, 394, 405, 414, 421, 432, 438, 447, 459, 466, 477

Offset: 1

Clark Kimberling, Apr 01 2023

This is the fourth 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. Their limiting densities are 4/9, 2/9, 2/9, 1/9, respectively (and likewise for A360394-A360401).

			(1)  v o u = (3, 7, 10, 11, 14, 16, 17, 20, 23, 25, 26, 29, 30, 33, 37, ...) = A360402
(2)  v' o u = (1, 4, 9, 13, 19, 22, 24, 31, 36, 40, 42, 49, 51, 58, 64, ...) = A360403
(3)  v o u' = (5, 8, 12, 18, 21, 28, 32, 35, 39, 46, 50, 53, 59, 62, 67, ...) = A360404
(4)  v' o u' = (2, 6, 15, 27, 34, 45, 55, 60, 69, 81, 91, 96, 108, 114, ...) = A360405

Cf. A026530, A360392, A360393, A360394-A3546352 (intersections instead of results of compositions), A360398-A360401 (results of reversed compositions), A360402, A360403, A360404.

z = 2000; zz = 100;
u = Accumulate[1 + ThueMorse /@ Range[0, 600]]; (* A026430 *)
u1 = Complement[Range[Max[u]], u];  (* A356133 *)
v = u + 2;  (* A360392 *)
v1 = Complement[Range[Max[v]], v]; (* A360393 *)
Table[v[[u[[n]]]], {n, 1, zz}]     (* A360402 *)
Table[v1[[u[[n]]]], {n, 1, zz}]    (* A360403 *)
Table[v[[u1[[n]]]], {n, 1, zz}]    (* A360404 *)
Table[v1[[u1[[n]]]], {n, 1, zz}]   (* A360405 *)

A360394 Intersection of A026430 and A360392.

Original entry on oeis.org

3, 5, 8, 10, 12, 14, 16, 18, 21, 23, 26, 28, 30, 33, 35, 37, 39, 41, 44, 46, 48, 50, 52, 54, 57, 59, 61, 63, 65, 68, 70, 72, 75, 77, 80, 82, 84, 86, 88, 90, 93, 95, 98, 100, 102, 105, 107, 109, 111, 113, 116, 118, 120, 123, 125, 128, 130, 132, 134, 136, 138

Offset: 1

Clark Kimberling, Feb 05 2023

This is the first of four sequences that partition the positive integers. 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. The limiting densities of these four sequences are 4/9, 2/9, 2/9, and 1/9, respectively (and likewise for A360402-A360405).
For A360394, u, v, u', v', are sequences obtained from the Thue-Morse sequence, A026430, as follows:
u = A026530 = (1,3,5,6,8,9,10, 12, ... ) = partial sums of A026430
u' = A356133 = (2,4,7,11,13,17, 20, ... ) = complement of u
v = u + 1 = A285954, except its initial 1
v' = complement of v.

			(1)  u ^ v = (3, 5, 8, 10, 12, 14, 16, 18, 21, 23, 26, 28, 30, 33, ...) =  A360394
(2)  u ^ v' = (1, 6, 9, 15, 19, 24, 27, 31, 36, 42, 45, 51, 55, 60, ...) =  A360395
(3)  u' ^ v = (7, 11, 17, 20, 25, 29, 32, 38, 43, 47, 53, 56, 62, ...) = A360396
(4)  u' ^ v' = (2, 4, 13, 22, 34, 40, 49, 58, 64, 76, 85, 94, 106, ...) = A360397

Cf. A026430, A356133, A360392, A360393, A356850-A360405.

z = 400;
u = Accumulate[1 + ThueMorse /@ Range[0, z]];   (* A026430 *)
u1 = Complement[Range[Max[u]], u];  (* A356133 *)
v = u + 2 ; (* A360392 *)
v1 = Complement[Range[Max[v]], v];  (* A360393 *)
Intersection[u, v]     (* A360394 *)
Intersection[u, v1]    (* A360395 *)
Intersection[u1, v]    (* A360396 *)
Intersection[u1, v1]   (* A360397 *)

A359277 Intersection of A026430 and (1 + A285953).

Original entry on oeis.org

6, 9, 10, 15, 16, 19, 24, 27, 28, 31, 36, 37, 42, 45, 46, 51, 52, 55, 60, 61, 66, 69, 70, 73, 78, 81, 82, 87, 88, 91, 96, 99, 100, 103, 108, 109, 114, 117, 118, 121, 126, 129, 130, 135, 136, 139, 144, 145, 150, 153, 154, 159, 160, 163, 168, 171, 172, 175

Offset: 1

Clark Kimberling, Jan 26 2023

This is the first of three sequences that partition the positive integers. Taking u = A026430 and v = 1 + A285953 (which is A285953 except for its initial 1), the three sequences are (1) u ^ v = intersection of u and v (in increasing order); (2) u ^ v'; and (3) u' ^ v. The limiting density of each of these is 1/3.

			(1)  u ^ v = (6, 9, 10, 15, 16, 19, 24, 27, 28, 31, 36, 37, ...) =    A359277
(2)  u ^ v' = (1, 3, 5, 8, 12, 14, 18, 21, 23, 26, 30, 33, 35, ...) =  A285953, except for the initial 1
(3)  u' ^ v = (2, 4, 7, 11, 13, 17, 20, 22, 25, 29, 32, 34, 38, ...) = A356133

Cf. A026530, A285954, A356133, A359352 to A360139) (results of compositions instead of intersections).

z = 200;
u = Accumulate[1 + ThueMorse /@ Range[0, z]]   (* A026430 *)
u1 = Complement[Range[Max[u]], u]  (* A356133 *)
v = u + 1
v1 = Complement[Range[Max[v]], v]
Intersection[u, v]    (* A359277 *)
Intersection[u, v1]   (* A285953 *)
Intersection[u1, v]   (* A356133 *)

A359352 a(n) = A026430(1 + A026430(n)).

Original entry on oeis.org

3, 6, 9, 10, 14, 15, 16, 19, 23, 24, 26, 28, 30, 33, 36, 37, 41, 42, 44, 46, 48, 51, 54, 55, 57, 60, 63, 65, 68, 69, 70, 73, 77, 78, 80, 82, 84, 87, 90, 91, 93, 96, 99, 100, 103, 105, 107, 109, 111, 114, 117, 118, 121, 123, 125, 128, 130, 132, 134, 136, 138

Offset: 1

Clark Kimberling, Jan 26 2023

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) v' o u'.
Every positive integer is in exactly one of the four sequences. Their limiting densities are 4/9, 2/9, 2/9, 1/9, respectively.

			(1) u o v = (3, 6, 9, 10, 14, 15, 16, 19, 23, 24, 26, 28, 30, 33, 36, 37, 41, ...) = A359352
(2) u o v' = (1, 5, 8, 12, 18, 21, 27, 31, 35, 39, 45, 50, 52, 59, 61, 66, 72, ...) = A359353
(3) u' o v = (4, 11, 17, 20, 25, 29, 32, 38, 43, 47, 49, 56, 58, 64, 71, 74, ...) = A360134
(4) u' o v' = (2, 7, 13, 22, 34, 40, 53, 62, 67, 76, 89, 97, 104, 115, 122, ...) = A360135

Cf. A026530, A359352, A285953, A285954, A359277 (intersections instead of results of composition), A359353-A360139.

z = 2000; zz = 100;
u = Accumulate[1 + ThueMorse /@ Range[0, 600]]; (* A026430 *)
u1 = Complement[Range[Max[u]], u];  (* A356133 *)
v = u + 1;  (* A285954 *)
v1 = Complement[Range[Max[v]], v]; (* A285953 *)
Table[u[[v[[n]]]], {n, 1, zz}]     (* A359352 *)
Table[u[[v1[[n]]]], {n, 1, zz}]    (* A359353 *)
Table[u1[[v[[n]]]], {n, 1, zz}]    (* A360134 *)
Table[u1[[v1[[n]]]], {n, 1, zz}]   (* A360135 *)

def A359352(n): return (m:=n+1+(n-1>>1)+(n-1&1|(n.bit_count()&1^1)))+(m-1>>1)+(m-1&1|(m.bit_count()&1^1)) # Chai Wah Wu, Mar 01 2023

A360139 a(n) = V(A356133(n)), where V(1) = 1 and V(k) = A285953(k+1) for k >= 2.

Original entry on oeis.org

3, 8, 18, 30, 35, 48, 57, 63, 72, 84, 93, 98, 111, 116, 125, 138, 143, 156, 165, 170, 183, 188, 198, 209, 219, 224, 234, 245, 252, 263, 273, 279, 288, 300, 309, 314, 327, 332, 342, 353, 363, 368, 378, 390, 395, 408, 416, 422, 435, 440, 450, 462, 467, 480

Offset: 1

Clark Kimberling, Feb 03 2023

This is the fourth 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. Their limiting densities are 4/9, 2/9, 2/9, 1/9, respectively.

			(1)  v o u = (2, 6, 9, 10, 13, 15, 16, 19, 22, 24, 25, 28, 29, 32, 36, ...) = A360136
(2)  v' o u = (1, 5, 12, 14, 21, 23, 26, 33, 39, 41, 44, 50, 54, 59, 65, ...) = A360137
(3)  v o u' = (4, 7, 11, 17, 20, 27, 31, 34, 38, 45, 49, 52, 58, 61, 66, ...) = A360138
(4)  v' o u' = (3, 8, 18, 30, 35, 48, 57, 63, 72, 84, 93, 98, 111, 116, ...) = A360139

Cf. A026530, A359352, A285953, A359277 (intersections instead of results of composition), A359352-A360138.

z = 2000; zz = 100;
u = Accumulate[1 + ThueMorse /@ Range[0, 600]]; (* A026430 *)
u1 = Complement[Range[Max[u]], u];  (* A356133 *)
v = u + 1;  (* A285954 *)
v1 = Complement[Range[Max[v]], v];  (* A285953 *)
Table[v[[u[[n]]]], {n, 1, zz}]      (* A360136 *)
Table[v1[[u[[n]]]], {n, 1, zz}]     (* A360137 *)
Table[v[[u1[[n]]]], {n, 1, zz}]     (* A360138 *)
Table[v1[[u1[[n]]]], {n, 1, zz}]    (* A360139 *)

A360398 a(n) = A026430(1 + A360392(n)).

Original entry on oeis.org

5, 8, 10, 12, 15, 16, 18, 21, 24, 26, 27, 30, 31, 35, 37, 39, 42, 44, 45, 48, 50, 52, 55, 57, 59, 61, 65, 66, 69, 70, 72, 75, 78, 80, 81, 84, 86, 88, 91, 93, 95, 98, 100, 102, 105, 107, 108, 111, 113, 116, 118, 120, 123, 125, 126, 129, 132, 134, 135, 138

Offset: 1

Clark Kimberling, Feb 10 2023

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) v' o u'.
Every positive integer is in exactly one of the four sequences. Their limiting densities are 4/9, 2/9, 2/9, 1/9 (and likewise for A360394-A360397 and A360402-A360405).

			(1)  u o v = (5, 8, 10, 12, 15, 16, 18, 21, 24, 26, 27, 30, 31, 35, 37, 39, ...) = A360398
(2)  u o v' = (1, 3, 6, 9, 14, 19, 23, 28, 33, 36, 41, 46, 51, 54, 60, 63, 68, ...) = A360399
(3)  u' o v = (7, 13, 20, 22, 29, 32, 34, 40, 47, 49, 53, 58, 62, 67, 74, 76, ...) = A360400
(4)  u' o v' = (2, 4, 11, 17, 25, 38, 43, 56, 64, 71, 79, 92, 101, 106, 119, ...) = A360401

Cf. A026530, A356133, A360392, A360393, A360399, A286355, A286356, A360394 (intersections instead of results of composition), A360402-A360405.

z = 2000;
u = Accumulate[1 + ThueMorse /@ Range[0, 600]]; (* A026430 *)
u1 = Complement[Range[Max[u]], u];  (* A356133 *)
v = u + 2;  (* A360392 *)
v1 = Complement[Range[Max[v]], v];    (* A360393 *)
zz = 100;
Table[u[[v[[n]]]], {n, 1, zz}]    (* A360398 *)
Table[u[[v1[[n]]]], {n, 1, zz}]   (* A360399 *)
Table[u1[[v[[n]]]], {n, 1, zz}]   (* A360400 *)
Table[u1[[v1[[n]]]], {n, 1, zz}]  (* A360401 *)

A360402 a(n) = A360392(A026430(n)).

Original entry on oeis.org

3, 7, 10, 11, 14, 16, 17, 20, 23, 25, 26, 29, 30, 33, 37, 38, 41, 43, 44, 47, 48, 52, 54, 56, 57, 61, 63, 65, 68, 70, 71, 74, 77, 79, 80, 83, 84, 88, 90, 92, 93, 97, 100, 101, 104, 105, 107, 110, 111, 115, 118, 119, 122, 123, 125, 128, 131, 132, 134, 137

Offset: 1

Clark Kimberling, Mar 11 2023

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. Their limiting densities are 4/9, 2/9, 2/9, 1/9, respectively (and likewise for A360394-A360401).

			(1)  v o u = (3, 7, 10, 11, 14, 16, 17, 20, 23, 25, 26, 29, 30, 33, 37, ...) = A360402
(2)  v' o u = (1, 4, 9, 13, 19, 22, 24, 31, 36, 40, 42, 49, 51, 58, 64, ...) = A360403
(3)  v o u' = (5, 8, 12, 18, 21, 28, 32, 35, 39, 46, 50, 53, 59, 62, 67, ...) = A360404
(4)  v' o u' = (2, 6, 15, 27, 34, 45, 55, 60, 69, 81, 91, 96, 108, 114, ...) = A360405

Winston de Greef, Table of n, a(n) for n = 1..10000

Cf. A026530, A360392, A360393, A360394-A3546352 (intersections instead of results of compositions), A360398-A360401 (results of reversed compositions), A360403, A360404, A360405.

z = 2000; zz = 100;
u = Accumulate[1 + ThueMorse /@ Range[0, 600]]; (* A026430 *)
u1 = Complement[Range[Max[u]], u];  (* A356133 *)
v = u + 2;  (* A360392 *)
v1 = Complement[Range[Max[v]], v];  (* A360393 *)
Table[v[[u[[n]]]], {n, 1, zz}]    (* A360402 *)
Table[v1[[u[[n]]]], {n, 1, zz}]   (* A360403 *)
Table[v[[u1[[n]]]], {n, 1, zz}]   (* A360404 *)
Table[v1[[u1[[n]]]], {n, 1, zz}]  (* A360405 *)

def A360392(n): return n+2+(n-1>>1)+(n-1&1|(n.bit_count()&1^1))
def A026430(n): return n+(n-1>>1)+(n-1&1|(n.bit_count()&1^1))
def A360402(n): return A360392(A026430(n)) # Winston de Greef, Mar 24 2023

A360397 Intersection of A356133 and A360393.

Original entry on oeis.org

2, 4, 13, 22, 34, 40, 49, 58, 64, 76, 85, 94, 106, 112, 124, 133, 142, 148, 157, 166, 178, 184, 193, 202, 208, 220, 229, 238, 244, 253, 262, 274, 280, 292, 301, 310, 322, 328, 337, 346, 352, 364, 373, 382, 394, 400, 412, 421, 430, 436, 445, 454, 466, 472

Offset: 1

Clark Kimberling, Feb 10 2023

This is the fourth of four sequences that partition the positive integers. 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. The limiting densities of these four sequences are 4/9, 2/9, 2/9, and 1/9, respectively.
For A360397, u, v, u', v', are sequences obtained from the Thue-Morse sequence, A026430, as follows:
u = A026530 = (1,3,5,6,8,9,10, 12, ... ) = partial sums of A026430;
u' = A356133 = (2,4,7,11,13,17, 20, ... ) = complement of u;
v = u + 1 = A285954, except its initial 1;
v' = complement of v.

			(1)  u ^ v = (3, 5, 8, 10, 12, 14, 16, 18, 21, 23, 26, 28, 30, 33, ...) =  A360394
(2)  u ^ v' = (1, 6, 9, 15, 19, 24, 27, 31, 36, 42, 45, 51, 55, 60, ...) =  A360395
(3)  u' ^ v = (7, 11, 17, 20, 25, 29, 32, 38, 43, 47, 53, 56, 62, ...) = A360396
(4)  u' ^ v' = (2, 4, 13, 22, 34, 40, 49, 58, 64, 76, 85, 94, 106, ...) = A360397

Cf. A026430, A356133, A360392-A360396, A360398-A360405.

z = 400;
u = Accumulate[1 + ThueMorse /@ Range[0, z]];   (* A026430 *)
u1 = Complement[Range[Max[u]], u];  (* A356133 *)
v = u + 2 ; (* A360392 *)
v1 = Complement[Range[Max[v]], v];    (* A360393 *)
Intersection[u, v]    (* A360394 *)
Intersection[u, v1]   (* A360395 *)
Intersection[u1, v]   (* A360396 *)
Intersection[u1, v1]  (* A360397 *)

A360401 a(n) = A356133(A360393(n)).

Original entry on oeis.org

2, 4, 11, 17, 25, 38, 43, 56, 64, 71, 79, 92, 101, 106, 119, 124, 133, 146, 151, 164, 173, 178, 191, 197, 206, 218, 227, 233, 242, 253, 260, 272, 280, 287, 295, 308, 317, 322, 335, 341, 350, 362, 371, 377, 385, 398, 403, 415, 425, 430, 443, 449, 457, 470

Offset: 1

Clark Kimberling, Mar 11 2023

This is the third 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) v' o u'.
Every positive integer is in exactly one of the four sequences. Their limiting densities are 4/9, 2/9, 2/9, 1/9 (and likewise for A360394-A360397 and A360402-A360405).

			(1)  u o v = (5, 8, 10, 12, 15, 16, 18, 21, 24, 26, 27, 30, 31, 35, 37, 39, ...) = A360398
(2)  u o v' = (1, 3, 6, 9, 14, 19, 23, 28, 33, 36, 41, 46, 51, 54, 60, 63, 68, ...) = A360399
(3)  u' o v = (7, 13, 20, 22, 29, 32, 34, 40, 47, 49, 53, 58, 62, 67, 74, 76, ...) = A360400
(4)  u' o v' = (2, 4, 11, 17, 25, 38, 43, 56, 64, 71, 79, 92, 101, 106, 119, ...) = A360401

Cf. A026530, A356133, A360392, A360393, A360398, A286354, A286355, A360394 (intersections instead of results of composition), A360402-A360405.

z = 2000;
u = Accumulate[1 + ThueMorse /@ Range[0, 600]]; (* A026430 *)
u1 = Complement[Range[Max[u]], u];  (* A356133 *)
v = u + 2;  (* A360392 *)
v1 = Complement[Range[Max[v]], v];  (* A360393 *)
zz = 100;
Table[u[[v[[n]]]], {n, 1, zz}]    (* A360398 *)
Table[u[[v1[[n]]]], {n, 1, zz}]   (* A360399 *)
Table[u1[[v[[n]]]], {n, 1, zz}]   (* A360400 *)
Table[u1[[v1[[n]]]], {n, 1, zz}]  (* A360401 *)

A360135 a(n) = A356133(A285953(n+1)).

Original entry on oeis.org

2, 7, 13, 22, 34, 40, 53, 62, 67, 76, 89, 97, 104, 115, 122, 131, 142, 148, 161, 169, 176, 187, 193, 202, 215, 223, 229, 238, 251, 257, 269, 278, 283, 292, 305, 313, 320, 331, 337, 346, 359, 367, 373, 382, 394, 400, 412, 421, 428, 439, 445, 454, 466, 472

Offset: 1

Clark Kimberling, Jan 30 2023

This is the fourth 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) v' o u'.
Every positive integer is in exactly one of the four sequences. Their limiting densities are 4/9, 2/9, 2/9, 1/9, respectively.

			(1) u o v = (3, 6, 9, 10, 14, 15, 16, 19, 23, 24, 26, 28, 30, 33, 36, 37, 41, ...) = A359352
(2) u o v' = (1, 5, 8, 12, 18, 21, 27, 31, 35, 39, 45, 50, 52, 59, 61, 66, 72, ...) = A359353
(3) u' o v = (4, 11, 17, 20, 25, 29, 32, 38, 43, 47, 49, 56, 58, 64, 71, 74, ...) = A360134
(4) u' o v' = (2, 7, 13, 22, 34, 40, 53, 62, 67, 76, 89, 97, 104, 115, 122, ...) = A360135

Cf. A026530, A359352, A285953, A285954, A359277 (intersections instead of results of composition), A359352-A360134, A360136-A360139.

z = 2000; zz = 100;
u = Accumulate[1 + ThueMorse /@ Range[0, 600]]; (* A026430 *)
u1 = Complement[Range[Max[u]], u];  (* A356133 *)
v = u + 1;  (* A285954 *)
v1 = Complement[Range[Max[v]], v];  (* A285953 *)
Table[u[[v[[n]]]], {n, 1, zz}]      (* A359352 *)
Table[u[[v1[[n]]]], {n, 1, zz}]     (* A359353 *)
Table[u1[[v[[n]]]], {n, 1, zz}]     (* A360134 *)
Table[u1[[v1[[n]]]], {n, 1, zz}]    (* A360135 *)

cp's OEIS Frontend

A360405 a(n) = A360393(A356133(n)).

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A360394 Intersection of A026430 and A360392.

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A359277 Intersection of A026430 and (1 + A285953).

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A359352 a(n) = A026430(1 + A026430(n)).

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A360139 a(n) = V(A356133(n)), where V(1) = 1 and V(k) = A285953(k+1) for k >= 2.

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A360398 a(n) = A026430(1 + A360392(n)).

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A360402 a(n) = A360392(A026430(n)).

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A360397 Intersection of A356133 and A360393.

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Mathematica

A360401 a(n) = A356133(A360393(n)).

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