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

1,2

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

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 *)

Prog

(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

Crossrefs

Cf. A007913, A156595, A189716, A189717.

Union of A055040 and A055047.

Sequence in context: A080746 A246362 A069909 * A101993 A370267 A002481

Adjacent sequences: A189712 A189713 A189714 * A189716 A189717 A189718

Keyword

nonn,changed

Author

Clark Kimberling, Apr 26 2011

Extensions

Name enhanced by Peter Munn, May 17 2020

Status

approved