A189715 Numbers k such that A156595(k-1) = 0; complement of A189716.

1, 4, 6, 7, 9, 10, 13, 15, 16, 19, 22, 24, 25, 28, 31, 33, 34, 36, 37, 40, 42, 43, 46, 49, 51, 52, 54, 55, 58, 60, 61, 63, 64, 67, 69, 70, 73, 76, 78, 79, 81, 82, 85, 87, 88, 90, 91, 94, 96, 97, 100, 103, 105, 106, 109, 112, 114, 115, 117, 118, 121, 123, 124, 127, 130, 132, 133, 135, 136, 139, 141, 142, 144, 145, 148, 150, 151, 154, 157, 159

Offset: 1

Author

Clark Kimberling, Apr 26 2011

Keywords

• nonn

Comments

See A156595.

Numbers whose squarefree part is congruent modulo 9 to 1, 4, 6 or 7. - Peter Munn, May 17 2020

The asymptotic density of this sequence is 1/2. - Amiram Eldar, Mar 08 2021

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A007913, A156595, A189716, A189717.

Union of A055040 and A055047.

Programs

Mathematica

t = Nest[Flatten[# /. {0->{0,1,1}, 1->{0,1,0}}] &, {0}, 5] (*A156595*)
f[n_] := t[[n]]
Flatten[Position[t, 0]] (*A189715*)
Flatten[Position[t, 1]] (*A189716*)
s[n_] := Sum[f[i], {i, 1, n}]; s[0] = 0;
Table[s[n], {n, 1, 120}] (*A189717*)
f[p_, e_] := (p^Mod[e, 2]); sqfpart[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[160], MemberQ[{1, 4, 6, 7}, Mod[sqfpart[#], 9]] &] (* Amiram Eldar, Mar 08 2021 *)

Python

from sympy import integer_log
def A189715(n):
    def f(x): return n+x-sum(((m:=x//9**i)-1)//9+(m-4)//9+(m-6)//9+(m-7)//9+4 for i in range(integer_log(x,9)[0]+1))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Feb 15 2025

Extensions

Name enhanced by Peter Munn, May 17 2020