A145397 Numbers not of the form m*(m+1)*(m+2)/6, the non-tetrahedral numbers.

2, 3, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78

Offset: 1

Author

Reinhard Zumkeller, Oct 14 2008

Keywords

• nonn

Comments

Complement of A000292; A000040 is a subsequence.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..5000
• Cristinel Mortici, Remarks on Complementary Sequences, Fibonacci Quart. 48 (2010), no. 4, 343-347.

Crossrefs

Cf. A000040, A000292, A014306, A023533.

Programs

Magma

[n: n in [1..100] | Binomial(Floor((6*n-1)^(1/3))+2, 3) ne n ]; // G. C. Greubel, Feb 20 2022

Mathematica

Select[Range[100], Binomial[Floor[Surd[6*# -1, 3]] +2, 3] != # &] (* G. C. Greubel, Feb 20 2022 *)

PARI

is(n)=binomial(sqrtnint(6*n,3)+2,3)!=n \\ Charles R Greathouse IV, Feb 22 2017

Python

from itertools import count
from math import comb
from sympy import integer_nthroot
def A145397(n):
    def f(x): return n+next(i for i in count(integer_nthroot(6*x,3)[0],-1) if comb(i+2,3)<=x)
    def iterfun(f,n=0):
        m, k = n, f(n)
        while m != k: m, k = k, f(k)
        return m
    return iterfun(f,n) # Chai Wah Wu, Sep 09 2024

Python

from math import comb
from sympy import integer_nthroot
def A145397(n): return n+(m:=integer_nthroot(6*n,3)[0])-(n+m<=comb(m+2,3)) # Chai Wah Wu, Oct 01 2024

Sage

[n for n in (1..100) if binomial( floor( real_nth_root(6*n-1, 3) ) +2, 3) != n ] # G. C. Greubel, Feb 20 2022

Formula

A014306(a(n)) = 1; A023533(a(n)) = 0.

a(n) = n+m if 6(n+m)>m(m+1)(m+2) and a(n)=n+m-1 otherwise where m is floor((6n)^(1/3)). - Chai Wah Wu, Oct 01 2024

Extensions

Definition corrected by Ant King, Sep 20 2012