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).

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

Author

Clark Kimberling, Jan 13 2025

Keywords

• nonn

Comments

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...

Crossrefs

Cf. A000027, A184812, A378142, A378185.

Programs

Mathematica

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 *)

Formula

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