A136803 -id:A136803 - cp's OEIS frontend

A356056 a(n) = A001951(A137803(n)).

1, 4, 7, 9, 12, 15, 18, 21, 24, 26, 29, 31, 33, 36, 39, 42, 45, 48, 50, 53, 56, 59, 62, 63, 66, 69, 72, 74, 77, 80, 83, 86, 89, 91, 93, 96, 98, 101, 104, 107, 110, 113, 115, 118, 121, 124, 125, 128, 131, 134, 137, 140, 142, 145, 148, 151, 154, 156, 158, 161

Offset: 1

Clark Kimberling, Jul 26 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.
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 A356056, u, v, u', v', are the Beatty sequences given by u(n) = floor(n*sqrt(2)) and v(n) = floor(n*(1/2 + sqrt(2))), so that r = sqrt(2), s = 1/2 + sqrt(2), r' = 2 + sqrt(2), s' = (9 + 4*sqrt(2))/7.

			(1)  u o v = (1, 4, 7, 9, 12, 15, 18, 21, 24, 26, 29, 31, ...) = A356056
(2)  u o v' = (2, 5, 8, 11, 14, 16, 19, 22, 25, 28, 32, 35, ...) = A356057
(3)  u' o v = (3, 10, 17, 23, 30, 37, 44, 51, 58, 64, 71, ...) = A356058
(4)  u' o v' = (6, 13, 20, 27, 34, 40, 47, 54, 61, 68, 78, ...) = A356059

Cf. A001951, A001952, A136803, A137804, A356052 (intersections instead of results of composition), A356057, A356058, A356059.

z = 800;
u = Table[Floor[n (Sqrt[2])], {n, 1, z}]   (* A001951 *)
u1 = Complement[Range[Max[u]], u]     (* A001952 *)
v = Table[Floor[n (1/2 + Sqrt[2])], {n, 1, z}]  (* A137803 *)
v1 = Complement[Range[Max[v]], v]  (* A137804 *)
Table[u[[v[[n]]]], {n, 1, z/8}];   (* A356056 *)
Table[u[[v1[[n]]]], {n, 1, z/8}];  (* A356057 *)
Table[u1[[v[[n]]]], {n, 1, z/8}];  (* A356058 *)
Table[u1[[v1[[n]]]], {n, 1, z/8}]; (* A356059 *)

a(n) = A001951(A137803(n)).

A356052 Intersection of A001951 and A137803.

Original entry on oeis.org

1, 5, 7, 9, 11, 15, 19, 21, 22, 24, 26, 28, 32, 36, 38, 42, 45, 49, 53, 55, 57, 59, 63, 65, 66, 70, 72, 74, 76, 80, 82, 84, 86, 89, 91, 93, 97, 101, 103, 107, 111, 114, 118, 120, 124, 128, 130, 132, 135, 137, 141, 145, 147, 149, 151, 155, 156, 158, 162, 164

Offset: 1

Clark Kimberling, Jul 26 2022

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.
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 A356052, u, v, u', v', are the Beatty sequences given by u(n) = floor(n*sqrt(2)) and v(n) = floor(n*(1/2 + sqrt(2))), so that r = sqrt(2), s = 1/2 + sqrt(2), r' = 2 + sqrt(2), s' = (9 + 4*sqrt(2))/7.

			(1)  u ^ v = (1, 5, 7, 9, 11, 15, 19, 21, 22, 24, 26, 28, ...) = A356052
(2)  u ^ v' = (2, 4, 8, 12, 14, 16, 18, 25, 29, 31, 33, 35, ...) = A356053
(3)  u' ^ v = (3, 13, 17, 30, 34, 40, 44, 47, 51, 61, 68, ...) = A356054
(4)  u' ^ v' = (6, 10, 20, 23, 27, 37, 54, 58, 64, 71, 75, ...) = A356055

Cf. A001951, A001952, A136803, A137804, A356053, A356054, A356055, A356056 (composites instead of intersections), A356081.

z = 250;
u = Table[Floor[n (Sqrt[2])], {n, 1, z}]   (* A001951 *)
u1 = Complement[Range[Max[u]], u]     (* A001952 *)
v = Table[Floor[n (1/2 + Sqrt[2])], {n, 1, z}]  (* A137803 *)
v1 = Complement[Range[Max[v]], v]     (* A137804 *)
Intersection[u, v]    (* A356052 *)
Intersection[u, v1]   (* A356053 *)
Intersection[u1, v]   (* A356054 *)
Intersection[u1, v1]  (* A356055 *)

A356057 a(n) = A001951(A137804(n)).

Original entry on oeis.org

2, 5, 8, 11, 14, 16, 19, 22, 25, 28, 32, 35, 38, 41, 43, 46, 49, 52, 55, 57, 60, 65, 67, 70, 73, 76, 79, 82, 84, 87, 90, 94, 97, 100, 103, 106, 108, 111, 114, 117, 120, 123, 127, 130, 132, 135, 138, 141, 144, 147, 149, 152, 155, 159, 162, 165, 168, 171, 173

Offset: 1

Clark Kimberling, Jul 26 2022

This is the second of four sequences that partition the positive integers. See A356056.

			(1)  u o v = (1, 4, 7, 9, 12, 15, 18, 21, 24, 26, 29, 31, ...) = A356056
(2)  u o v' = (2, 5, 8, 11, 14, 16, 19, 22, 25, 28, 32, 35, ...) = A356057
(3)  u' o v = (3, 10, 17, 23, 30, 37, 44, 51, 58, 64, 71, ...) = A356058
(4)  u' o v' = (6, 13, 20, 27, 34, 40, 47, 54, 61, 68, 78, ...) = A356059

Cf. A001951, A001952, A136803, A137804, A356052 (intersections instead of results of composition), A356056, A356058, A356059.

u = Table[Floor[n (Sqrt[2])], {n, 1, z}]   (* A001951 *)
u1 = Complement[Range[Max[u]], u]     (* A001952 *)
v = Table[Floor[n (1/2 + Sqrt[2])], {n, 1, z}]  (* A137803 *)
v1 = Complement[Range[Max[v]], v]  (* A137804 *)
Table[u[[v[[n]]]], {n, 1, z/8}];   (* A356056 *)
Table[u[[v1[[n]]]], {n, 1, z/8}];  (* A356057 *)
Table[u1[[v[[n]]]], {n, 1, z/8}];  (* A356058 *)
Table[u1[[v1[[n]]]], {n, 1, z/8}]; (* A356059 *)

a(n) = A001951(A137804(n)).

A356058 a(n) = A001952(A137803(n)).

Original entry on oeis.org

3, 10, 17, 23, 30, 37, 44, 51, 58, 64, 71, 75, 81, 88, 95, 102, 109, 116, 122, 129, 136, 143, 150, 153, 160, 167, 174, 180, 187, 194, 201, 208, 215, 221, 225, 232, 238, 245, 252, 259, 266, 273, 279, 286, 293, 300, 303, 310, 317, 324, 331, 338, 344, 351, 358

Offset: 1

Clark Kimberling, Jul 26 2022

This is the third of four sequences that partition the positive integers. See A356056.

			(1)  u o v   = (1,  4,  7,  9, 12, 15, 18, 21, 24, 26, 29, ...) = A356056
(2)  u o v'  = (2,  5,  8, 11, 14, 16, 19, 22, 25, 28, 32, ...) = A356057
(3)  u' o v  = (3, 10, 17, 23, 30, 37, 44, 51, 58, 64, 71, ...) = A356058
(4)  u' o v' = (6, 13, 20, 27, 34, 40, 47, 54, 61, 68, 78, ...) = A356059

Cf. A001951, A001952, A136803, A137804, A356052 (intersections instead of the results of composition), A356056, A356057, A356059.

u = Table[Floor[n (Sqrt[2])], {n, 1, z}]   (* A001951 *)
u1 = Complement[Range[Max[u]], u]     (* A001952 *)
v = Table[Floor[n (1/2 + Sqrt[2])], {n, 1, z}]  (* A137803 *)
v1 = Complement[Range[Max[v]], v]  (* A137804 *)
Table[u[[v[[n]]]], {n, 1, z/8}];   (* A356056 *)
Table[u[[v1[[n]]]], {n, 1, z/8}];  (* A356057 *)
Table[u1[[v[[n]]]], {n, 1, z/8}];  (* A356058 *)
Table[u1[[v1[[n]]]], {n, 1, z/8}]; (* A356059 *)

a(n) = A001952(A137803(n)).

A356059 a(n) = A001952(A137804(n)).

Original entry on oeis.org

6, 13, 20, 27, 34, 40, 47, 54, 61, 68, 78, 85, 92, 99, 105, 112, 119, 126, 133, 139, 146, 157, 163, 170, 177, 184, 191, 198, 204, 211, 218, 228, 235, 242, 249, 256, 262, 269, 276, 283, 290, 297, 307, 314, 320, 327, 334, 341, 348, 355, 361, 368, 375, 385, 392

Offset: 1

Clark Kimberling, Jul 26 2022

This is the fourth of four sequences that partition the positive integers. See A356056.

			(1)  u o v = (1, 4, 7, 9, 12, 15, 18, 21, 24, 26, 29, 31, ...) = A356056
(2)  u o v' = (2, 5, 8, 11, 14, 16, 19, 22, 25, 28, 32, 35, ...) = A356057
(3)  u' o v = (3, 10, 17, 23, 30, 37, 44, 51, 58, 64, 71, ...) = A356058
(4)  u' o v' = (6, 13, 20, 27, 34, 40, 47, 54, 61, 68, 78, ...) = A356059

Cf. A001951, A001952, A136803, A137804, A356052 (intersections instead of results of composition), A356056, A356057, A356058.

u = Table[Floor[n (Sqrt[2])], {n, 1, z}]   (* A001951 *)
u1 = Complement[Range[Max[u]], u]     (* A001952 *)
v = Table[Floor[n (1/2 + Sqrt[2])], {n, 1, z}]  (* A137803 *)
v1 = Complement[Range[Max[v]], v]  (* A137804 *)
Table[u[[v[[n]]]], {n, 1, z/8}];   (* A356056 *)
Table[u[[v1[[n]]]], {n, 1, z/8}];  (* A356057 *)
Table[u1[[v[[n]]]], {n, 1, z/8}];  (* A356058 *)
Table[u1[[v1[[n]]]], {n, 1, z/8}]; (* A356059 *)

a(n) = A001952(A137804(n)).

A136806 Nonsquares mod 65537.

Original entry on oeis.org

3, 5, 6, 7, 10, 11, 12, 14, 20, 22, 23, 24, 27, 28, 29, 31, 39, 40, 41, 43, 44, 45, 46, 47, 48, 51, 54, 56, 57, 58, 59, 61, 62, 63, 65, 67, 73, 75, 78, 80, 82, 83, 85, 86, 88, 89, 90, 91, 92, 94, 95, 96, 99, 101, 102, 105, 108, 111, 112, 113, 114, 116, 118, 119

Offset: 1

T. D. Noe, Jan 22 2008

Because 65537 is a Fermat prime, these numbers are all primitive roots (mod 65537). Complement of A136805.

			Since 7 is not a perfect square, and there are no solutions to x^2 = 7 mod 65537, 7 is in the sequence.
Although 8 is not a perfect square either, there are solutions to x^2 = 8 mod 65537, such as x = 8160, so 8 is not in the sequence.

Nathaniel Johnston, Table of n, a(n) for n = 1..32768 (full sequence)
OEIS Wiki, Index to sequences related to squares

Cf. A136805 (squares mod 65537); A136803 and A136804 ((non)squares mod 257).

Cf. A028730.

A136806 := {$(0..65536)}: for n from 0 to 65536 do A136806 := A136806 minus {n^2 mod 65537}: od: l:=sort(convert(A136806,list)): l[1..64]; # Nathaniel Johnston, Jun 23 2011
# Much more efficient:
S:= {$0..65536} minus {seq(i^2 mod 65537, i=0..65537/2)}:
A:= sort(convert(S,list)):
A[1..64]; # Robert Israel, Nov 15 2017

p = 65537; Select[Range[0, p - 1], JacobiSymbol[#, p] == -1 &]

A136806=select( is_A136806(n)=!issquare(Mod(n,65537)), [0..2^16]); \\ Strictly speaking, the is(.) function should include "&& n<65537" according to the intended meaning of the definition of this sequence. See A136804 for faster code, which would here cause a stack overflow for default settings. - M. F. Hasler, Nov 15 2017

(1 to 65537).diff(((1: BigInt) to (65537: BigInt)).map(n => n * n % 65537)) // Alonso del Arte, Jan 17 2020

a(n) + a(32769 - n) = 65537.

A136804 Nonsquares mod 257.

Original entry on oeis.org

3, 5, 6, 7, 10, 12, 14, 19, 20, 24, 27, 28, 33, 37, 38, 39, 40, 41, 43, 45, 47, 48, 51, 53, 54, 55, 56, 63, 65, 66, 69, 71, 74, 75, 76, 77, 78, 80, 82, 83, 85, 86, 87, 90, 91, 93, 94, 96, 97, 101, 102, 103, 105, 106, 107, 108, 109, 110, 112, 115, 119, 125, 126

Offset: 1

T. D. Noe, Jan 22 2008

Because 257 is a Fermat prime, these numbers are all primitive roots (mod 257). Complement of A136803.

Nathaniel Johnston, Table of n, a(n) for n = 1..128 (full sequence)
Index entries for sequences related to squares

Cf. A136803 (squares mod 257), A136805 and A136806 (squares/nonsquares mod 65537).

A136804 := {$(0..256)}: for n from 0 to 256 do A136804 := A136804 minus {n^2 mod 257}: od: op(sort(convert(A136804,list))); # Nathaniel Johnston, Jun 23 2011

p=257; Select[Range[0,p-1], JacobiSymbol[ #,p]==-1&]

for (n=0, 256, if (! issquare(Mod(n, 257)), print1(n, ", "))) \\ Michel Marcus, Mar 12 2017

A136804=setminus(S=[0..256],Set([k^2 | k <- S]%257)); \\ M. F. Hasler, Nov 15 2017

a(n) + a(129-n) = 257.

A136805 Squares mod 65537.

Original entry on oeis.org

0, 1, 2, 4, 8, 9, 13, 15, 16, 17, 18, 19, 21, 25, 26, 30, 32, 33, 34, 35, 36, 37, 38, 42, 49, 50, 52, 53, 55, 60, 64, 66, 68, 69, 70, 71, 72, 74, 76, 77, 79, 81, 84, 87, 93, 97, 98, 100, 103, 104, 106, 107, 109, 110, 115, 117, 120, 121, 123, 128, 129, 132, 135

Offset: 1

T. D. Noe, Jan 22 2008

Because 65537 is a Fermat prime, the complement of this set, A136806, is the set of primitive roots (mod 65537).

Nathaniel Johnston, Table of n, a(n) for n = 1..32769 (full sequence)
OEIS Wiki, Index to sequences related to squares

Cf. A136806 (nonsquares mod 65537); A136803 and A136804 ((non)squares mod 257).

Cf. A010379.

A136805:={}: for n from 0 to 65536 do A136805 := A136805 union {n^2 mod 65537}: od: l:=sort(convert(A136805,list)): l[1..63]; # Nathaniel Johnston, Jun 23 2011

p = 65537; Select[Range[0, p - 1], JacobiSymbol[#, p] == 1 &]

A136805=Set([k^2 | k <- [0..2^16]]%65537); \\ M. F. Hasler, Nov 15 2017

[quadratic_residues(65537)] # Zerinvary Lajos, May 24 2009

((1: BigInt) to (65537: BigInt)).map(n => (n * n) % 65537).toSet.toSeq.sorted // Alonso del Arte, Dec 17 2019

a(n) + a(32771 - n) = 65537 for n > 1.

A356053 Intersection of A001951 and A137804.

Original entry on oeis.org

2, 4, 8, 12, 14, 16, 18, 25, 29, 31, 33, 35, 39, 41, 43, 46, 48, 50, 52, 56, 60, 62, 67, 69, 73, 77, 79, 83, 87, 90, 94, 96, 98, 100, 104, 106, 108, 110, 113, 115, 117, 121, 123, 125, 127, 131, 134, 138, 140, 142, 144, 148, 152, 154, 159, 161, 165, 169, 171

Offset: 1

Clark Kimberling, Jul 26 2022

This is the second of four sequences, u^v, u^v', u'^v, u'^v', that partition the positive integers. See A356052.

			(1)  u ^ v = (1, 5, 7, 9, 11, 15, 19, 21, 22, 24, 26, 28, ...) =  A356052
(2)  u ^ v' = (2, 4, 8, 12, 14, 16, 18, 25, 29, 31, 33, 35, ...) =  A356053
(3)  u' ^ v = (3, 13, 17, 30, 34, 40, 44, 47, 51, 61, 68, ...) = A356054
(4)  u' ^ v' = (6, 10, 20, 23, 27, 37, 54, 58, 64, 71, 75, ...) = A356055

Cf. A001951, A001952, A136803, A137804, A356052, A356054, A356055, A356056 (composites instead of intersections).

z = 250;
u = Table[Floor[n (Sqrt[2])], {n, 1, z}]   (* A001951 *)
u1 = Complement[Range[Max[u]], u]     (* A001952 *)
v = Table[Floor[n (1/2 + Sqrt[2])], {n, 1, z}]  (* A137803 *)
v1 = Complement[Range[Max[v]], v]     (* A137804 *)
Intersection[u, v]    (* A356052 *)
Intersection[u, v1]   (* A356053 *)
Intersection[u1, v]   (* A356054 *)
Intersection[u1, v1]  (* A356055 *)

A356054 Intersection of A001952 and A137803.

Original entry on oeis.org

3, 13, 17, 30, 34, 40, 44, 47, 51, 61, 68, 78, 88, 95, 99, 105, 109, 112, 116, 122, 126, 133, 139, 143, 153, 160, 170, 174, 187, 191, 204, 208, 218, 225, 235, 245, 252, 256, 262, 266, 269, 273, 279, 283, 290, 300, 310, 317, 327, 331, 334, 338, 344, 348

Offset: 1

Clark Kimberling, Jul 26 2022

This is the third of four sequences, u^v, u^v', u'^v, u'^v', that partition the positive integers. See A356052.

			(1)  u ^ v = (1, 5, 7, 9, 11, 15, 19, 21, 22, 24, 26, 28, ...) =  A356052
(2)  u ^ v' = (2, 4, 8, 12, 14, 16, 18, 25, 29, 31, 33, 35, ...) =  A356053
(3)  u' ^ v = (3, 13, 17, 30, 34, 40, 44, 47, 51, 61, 68, ...) = A356054
(4)  u' ^ v' = (6, 10, 20, 23, 27, 37, 54, 58, 64, 71, 75, ...) = A356055

Cf. A001951, A001952, A136803, A137804, A356052, A356054, A356055, A356056 (composites instead of intersections).

z = 250;
u = Table[Floor[n (Sqrt[2])], {n, 1, z}]   (* A001951 *)
u1 = Complement[Range[Max[u]], u]     (* A001952 *)
v = Table[Floor[n (1/2 + Sqrt[2])], {n, 1, z}]  (* A137803 *)
v1 = Complement[Range[Max[v]], v]     (* A137804 *)
Intersection[u, v]    (* A356052 *)
Intersection[u, v1]   (* A356053 *)
Intersection[u1, v]   (* A356054 *)
Intersection[u1, v1]  (* A356055 *)

cp's OEIS Frontend

A356056 a(n) = A001951(A137803(n)).

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A356052 Intersection of A001951 and A137803.

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A356057 a(n) = A001951(A137804(n)).

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A356058 a(n) = A001952(A137803(n)).

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A356059 a(n) = A001952(A137804(n)).

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A136806 Nonsquares mod 65537.

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A136804 Nonsquares mod 257.

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A136805 Squares mod 65537.

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