A376573 Numbers that are not squares of triangular numbers. Complement of A000537.

2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70

Offset: 1

Author

Chai Wah Wu, Oct 02 2024

Keywords

• nonn
• easy

Comments

Numbers that are not the sum of the first k cubes.

Links

• Paolo Xausa, Table of n, a(n) for n = 1..11011
• Eric Weisstein's World of Mathematics, Triangular Number

Crossrefs

Cf. A000537, A302058.

Programs

Mathematica

A376573[n_] := With[{m = Floor[(4*n)^(1/4)]}, n + m - Boole[4*n <= m*(m - 1)*(m*(m + 3) + 4)]];
Array[A376573, 96] (* or *)
Complement[Range[Last[#]], #] & [Accumulate[Range[4]^3]] (* Paolo Xausa, Oct 04 2024 *)

PARI

isok(k) = !issquare(k) || (issquare(k) && !ispolygonal(sqrtint(k), 3)); \\ Michel Marcus, Oct 02 2024

Python

from sympy import integer_nthroot
def A376573(n): return n+(m:=integer_nthroot(k:=n<<2,4)[0])-(k<=m*(m-1)*(m**2+3*m+4))

Formula

a(n) = n+m if 4n>m*(m-1)*(m^2+3*m+4) and a(n) = n+m-1 otherwise where m = floor((4n)^(1/4)).