A184809 -id:A184809 - cp's OEIS frontend

A184808 n + floor(r*n), where r = sqrt(2/3); complement of A184809.

1, 3, 5, 7, 9, 10, 12, 14, 16, 18, 19, 21, 23, 25, 27, 29, 30, 32, 34, 36, 38, 39, 41, 43, 45, 47, 49, 50, 52, 54, 56, 58, 59, 61, 63, 65, 67, 69, 70, 72, 74, 76, 78, 79, 81, 83, 85, 87, 89, 90, 92, 94, 96, 98, 99, 101, 103, 105, 107, 108, 110, 112, 114, 116

Offset: 1

Clark Kimberling, Jan 22 2011

nonn

This is the Beatty sequence for 1 + sqrt(2/3).
Also, a(n) is the position of 2*n^2 in the sequence obtained by arranging all the numbers in the sets {2*h^2, h >= 1} and {3*k^2, k >= 1} in increasing order. - Clark Kimberling, Oct 20 2014
Also, numbers n such that floor((n+1)*sqrt(6)) - floor(n*sqrt(6)) = 2. - Clark Kimberling, Jul 15 2015

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A184809, A182760 (comment about joint ranking),

A184810, A184811, A249096.

[n+Floor(n*Sqrt(2/3)): n in [1..70]]; // Vincenzo Librandi, Oct 23 2014

r=(2/3)^(1/2); s=(3/2)^(1/2);
a[n_]:=n+Floor [n*r];
b[n_]:=n+Floor [n*s];
Table[a[n],{n,1,120}]  (* A184808 *)
Table[b[n],{n,1,120}]  (* A184809 *)

main(size)={return(vector(size, n, n+floor(sqrt(2/3)*n)))} /* Anders Hellström, Jul 15 2015 */

a(n) = n + floor(r*n), where r = sqrt(2/3).

A249096 {2h^2, h >=1} union {3k^2, k >=1}, in increasing order.

Original entry on oeis.org

2, 3, 8, 12, 18, 27, 32, 48, 50, 72, 75, 98, 108, 128, 147, 162, 192, 200, 242, 243, 288, 300, 338, 363, 392, 432, 450, 507, 512, 578, 588, 648, 675, 722, 768, 800, 867, 882, 968, 972, 1058, 1083, 1152, 1200, 1250, 1323, 1352, 1452, 1458, 1568, 1587, 1682

Offset: 1

Clark Kimberling, Oct 20 2014

Let S = {2*h^2, h >=1} and T = {3*k^2, k >=1}. Then S and T are disjoint. The position of 2*n^2 in (S union T) is A184808(n), and the position of 3*n^2 is A184809(n).

			{2*h^2, h >=1} = {2, 8, 18, 32, 50, 72, 98, 128, 162, 200, ...};
{3*k^2, k >=1} = {3, 12, 27, 48, 75, 108, 147, 192, 243, ...};
so the union is {2, 3, 8, 12, 18, 27, 32, 48, 50, 72, 75, ...}

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A184808, A184809.

A249367 is essentially the same sequence.

z = 120; s = Table[2 h^2, {h, 1, z}]; t = Table[3 k^2, {k, 1, z}]; v = Sort[Union[s, t]]

A184810 Numbers m such that prime(m) has the form floor(k*r), where r=sqrt(2/3); complement of A184811.

Original entry on oeis.org

2, 3, 4, 8, 9, 10, 13, 14, 15, 17, 18, 19, 22, 23, 24, 26, 27, 28, 31, 34, 38, 39, 41, 42, 45, 46, 48, 49, 52, 53, 55, 56, 59, 60, 61, 66, 68, 72, 75, 76, 78, 79, 81, 82, 85, 86, 88, 89, 90, 92, 95, 96, 98, 99, 100, 102, 103, 106, 108, 109, 110, 112, 113, 114, 116, 117, 119, 120, 121, 122, 123, 124, 126, 128, 130, 131, 134, 135, 137, 139, 141, 142, 146, 147, 148, 149, 151, 152, 156, 157, 159, 162, 164, 165, 167, 168, 169, 170, 171, 173, 174, 175, 176, 177, 180

Offset: 1

Clark Kimberling, Jan 22 2011

nonn

Cf. A184808, A184809, A184811.

r=(2/3)^(1/2);s=(3/2)^(1/2); (* complementary because of joint ranking of i*sqrt(2) and j*sqrt(3) *)
a[n_]:=n+Floor [n*r]; b[n_]:=n+Floor [n*s];
Table[a[n],{n,1,120}]  (* A184808 *)
Table[b[n],{n,1,120}]  (* A184809 *)
t1={};Do[If[PrimeQ[a[n]], AppendTo[t1,a[n]]],{n,1,600}]
t2={};Do[If[PrimeQ[a[n]], AppendTo[t2,n]],{n,1,600}]
t3={};Do[If[MemberQ[t1,Prime[n]],AppendTo[t3,n]],{n,1,300}];t3
t4={};Do[If[PrimeQ[b[n]], AppendTo[t4,b[n]]],{n,1,600}]
t5={};Do[If[PrimeQ[b[n]], AppendTo[t5,n]],{n,1,600}]
t6={};Do[If[MemberQ[t4,Prime[n]],AppendTo[t6,n]],{n,1,300}];t6
(* t3 and t6 match A184810 and A184811 *)

A184811 Numbers m such that prime(m) has the form floor(r*k), where r=sqrt(3/2); complement of A184810.

Original entry on oeis.org

1, 5, 6, 7, 11, 12, 16, 20, 21, 25, 29, 30, 32, 33, 35, 36, 37, 40, 43, 44, 47, 50, 51, 54, 57, 58, 62, 63, 64, 65, 67, 69, 70, 71, 73, 74, 77, 80, 83, 84, 87, 91, 93, 94, 97, 101, 104, 105, 107, 111, 115, 118, 125, 127, 129, 132, 133, 136, 138, 140, 143, 144, 145, 150, 153, 154, 155, 158, 160, 161, 163, 166, 172, 178, 179, 181, 185, 188, 192, 194, 197, 200, 205, 206, 207, 208, 211, 212, 213, 215, 216

Offset: 1

Clark Kimberling, Jan 22 2011

nonn

Cf. A184809, A184810.

Mathematica
```
(See A184810.)
```

cp's OEIS Frontend

A184808 n + floor(r*n), where r = sqrt(2/3); complement of A184809.

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A249096 {2h^2, h >=1} union {3k^2, k >=1}, in increasing order.

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A184810 Numbers m such that prime(m) has the form floor(k*r), where r=sqrt(2/3); complement of A184811.

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A184811 Numbers m such that prime(m) has the form floor(r*k), where r=sqrt(3/2); complement of A184810.

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cp's OEIS Frontend

A184808 n + floor(r*n), where r = sqrt(2/3); complement of A184809.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A249096 {2*h^2, h >=1} union {3*k^2, k >=1}, in increasing order.

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Author

Keywords

Comments

Examples

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Mathematica

A184810 Numbers m such that prime(m) has the form floor(k*r), where r=sqrt(2/3); complement of A184811.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

A184811 Numbers m such that prime(m) has the form floor(r*k), where r=sqrt(3/2); complement of A184810.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

A249096 {2h^2, h >=1} union {3k^2, k >=1}, in increasing order.