A378185 -id:A378185 - cp's OEIS frontend

A378142 a(n) = n + floor(ns/r) + floor(nt/r), where r=2^(1/4), s=2^(1/2), t=2^(3/4).

3, 6, 10, 13, 17, 21, 24, 28, 31, 35, 39, 42, 46, 49, 53, 57, 61, 64, 67, 71, 74, 79, 82, 85, 89, 92, 97, 100, 104, 107, 110, 115, 118, 122, 125, 128, 133, 136, 140, 143, 146, 150, 154, 158, 161, 165, 168, 172, 176, 179, 183, 186, 190, 194, 197, 201, 204

Offset: 1

Clark Kimberling, Jan 13 2025

nonn

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 integer k >= 1, 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...

Cf. A000027, A010767, A184812, A378185, A379510.

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 *)

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

Name corrected by Clark Kimberling, Jan 20 2025

A379510 a(n) = n + floor(nr/t) + floor(ns/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

Clark Kimberling, Jan 13 2025

nonn

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...

Cf. A000027, A184812, A378142, A378185.

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 *)

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

A184909 a(n) = n + floor(ns/r) + floor(nt/r), where r=2^(1/5), s=r^2, t=r^3.

Original entry on oeis.org

3, 6, 9, 13, 16, 19, 24, 27, 30, 34, 37, 40, 44, 48, 51, 55, 58, 61, 65, 68, 72, 76, 79, 82, 85, 89, 93, 96, 100, 103, 106, 110, 113, 117, 121, 124, 127, 131, 134, 137, 142, 145, 148, 152, 155, 158, 162, 166, 169, 172, 176, 179, 182, 187, 190, 193, 197, 200, 203, 207

Offset: 1

Clark Kimberling, Jan 25 2011

nonn

Cf. A005531, A005533, A011093.

Cf. A184910, A184911, A378142, A378185, A379510.

r = 2^(1/5); s = r^2; t = r^3;
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}]  (* A184909 *)
Table[b[n], {n, 1, 120}]  (* A184910 *)
Table[c[n], {n, 1, 120}]  (* A184911 *)
(* Clark Kimberling, Jan 18 2025 *)

Definition in name corrected by Clark Kimberling, Jan 18 2025

A184910 a(n) = n + floor(nr/s) + floor(nt/s) with r = 2^(1/5), s = r^2, t = r^3.

Original entry on oeis.org

2, 5, 8, 11, 14, 17, 21, 23, 26, 29, 32, 35, 38, 42, 45, 47, 50, 53, 56, 59, 63, 66, 69, 71, 74, 77, 81, 84, 87, 90, 92, 95, 98, 102, 105, 108, 111, 114, 116, 119, 123, 126, 129, 132, 135, 138, 140, 144, 147, 150, 153, 156, 159, 163, 165, 168, 171, 174, 177, 180, 184

Offset: 1

Clark Kimberling, Jan 25 2011

nonn

Cf. A184909, A184911, A378142, A378185, A379510.

Name correted by Alois P. Heinz, Jan 19 2025

A184911 a(n) = n + floor(nr/t) + floor(ns/t) with r = 2^(1/5), s = r^2, t = r^3.

Original entry on oeis.org

1, 4, 7, 10, 12, 15, 18, 20, 22, 25, 28, 31, 33, 36, 39, 41, 43, 46, 49, 52, 54, 57, 60, 62, 64, 67, 70, 73, 75, 78, 80, 83, 86, 88, 91, 94, 97, 99, 101, 104, 107, 109, 112, 115, 118, 120, 122, 125, 128, 130, 133, 136, 139, 141, 143, 146, 149, 151, 154, 157, 160, 161

Offset: 1

Clark Kimberling, Jan 25 2011

nonn

Cf. A184909, A184910, A378142, A378185, A379510.

Name corrected by Alois P. Heinz, Jan 19 2025

cp's OEIS Frontend

A378142 a(n) = n + floor(ns/r) + floor(nt/r), where r=2^(1/4), s=2^(1/2), t=2^(3/4).

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

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A184909 a(n) = n + floor(ns/r) + floor(nt/r), where r=2^(1/5), s=r^2, t=r^3.

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Author

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Programs

Mathematica

Extensions

A184910 a(n) = n + floor(nr/s) + floor(nt/s) with r = 2^(1/5), s = r^2, t = r^3.

Views

Author

Keywords

Crossrefs

Extensions

A184911 a(n) = n + floor(nr/t) + floor(ns/t) with r = 2^(1/5), s = r^2, t = r^3.

Views

Author

Keywords

Crossrefs

Extensions

cp's OEIS Frontend

A378142 a(n) = n + floor(n*s/r) + floor(n*t/r), where r=2^(1/4), s=2^(1/2), t=2^(3/4).

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Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

Extensions

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).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

A184909 a(n) = n + floor(n*s/r) + floor(n*t/r), where r=2^(1/5), s=r^2, t=r^3.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Extensions

A184910 a(n) = n + floor(n*r/s) + floor(n*t/s) with r = 2^(1/5), s = r^2, t = r^3.

Views

Author

Keywords

Crossrefs

Extensions

A184911 a(n) = n + floor(n*r/t) + floor(n*s/t) with r = 2^(1/5), s = r^2, t = r^3.

Views

Author

Keywords

Crossrefs

Extensions

A378142 a(n) = n + floor(ns/r) + floor(nt/r), where r=2^(1/4), s=2^(1/2), t=2^(3/4).

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

A184909 a(n) = n + floor(ns/r) + floor(nt/r), where r=2^(1/5), s=r^2, t=r^3.

A184910 a(n) = n + floor(nr/s) + floor(nt/s) with r = 2^(1/5), s = r^2, t = r^3.

A184911 a(n) = n + floor(nr/t) + floor(ns/t) with r = 2^(1/5), s = r^2, t = r^3.