A379414 a(n) = n + floor(n*s/r) + floor(n*t/r), where r = 3^(1/4), s = 3^(1/2), t = 3^(3/4).

3, 7, 11, 15, 19, 23, 28, 31, 35, 40, 44, 47, 52, 56, 59, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100, 105, 108, 112, 117, 120, 124, 129, 133, 136, 141, 145, 149, 153, 157, 161, 165, 169, 173, 177, 181, 185, 189, 194, 197, 201, 206, 210, 213, 218, 222, 225, 230

Offset: 1

Author

Clark Kimberling, Jan 18 2025

Keywords

• nonn

Comments

This sequence and A379415 and A379416 partition the positive integers; see A184812 for a proof.

For each k in A000027, write "a" if k=A379414(n) for some n, "b" if k=A379415(n) for some n, and "c" if k=A379416(n) for some n. Concatenating these letters for k = 1,2,3,... spells the following infinite word:

cbacbcabccabcbacbcacbcabcbcacbacbcabccbacbcabcacbcbacbcacbacbcbacbcacbcabcbaccbacbcabccabcbacbcacbcabcbcacbacbcabccbacbacbcabccbacbcabcacbcba

Crossrefs

Cf. A184812, A379415, A379416.

Cf. also A011002, A002194, A011022.

Programs

Mathematica

r = 3^(1/4); s = 3^(1/2); t = 3^(3/4);
Table[n + Floor[n*s/r] + Floor[n*t/r], {n, 1, 120}]  (* A379411 *)
Table[n + Floor[n*r/s] + Floor[n*t/s], {n, 1, 120}]  (* A379412 *)
Table[n + Floor[n*r/t] + Floor[n*s/t], {n, 1, 120}]  (* A379413 *)

Formula

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