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

%I A189715 #20 Feb 16 2025 01:11:16
%S A189715 1,4,6,7,9,10,13,15,16,19,22,24,25,28,31,33,34,36,37,40,42,43,46,49,
%T A189715 51,52,54,55,58,60,61,63,64,67,69,70,73,76,78,79,81,82,85,87,88,90,91,
%U A189715 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
%N A189715 Numbers k such that A156595(k-1) = 0; complement of A189716.
%C A189715 See A156595.
%C A189715 Numbers whose squarefree part is congruent modulo 9 to 1, 4, 6 or 7. - _Peter Munn_, May 17 2020
%C A189715 The asymptotic density of this sequence is 1/2. - _Amiram Eldar_, Mar 08 2021
%H A189715 Amiram Eldar, <a href="/A189715/b189715.txt">Table of n, a(n) for n = 1..10000</a>
%t A189715 t = Nest[Flatten[# /. {0->{0,1,1}, 1->{0,1,0}}] &, {0}, 5] (*A156595*)
%t A189715 f[n_] := t[[n]]
%t A189715 Flatten[Position[t, 0]] (*A189715*)
%t A189715 Flatten[Position[t, 1]] (*A189716*)
%t A189715 s[n_] := Sum[f[i], {i, 1, n}]; s[0] = 0;
%t A189715 Table[s[n], {n, 1, 120}] (*A189717*)
%t A189715 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 *)
%o A189715 (Python)
%o A189715 from sympy import integer_log
%o A189715 def A189715(n):
%o A189715     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))
%o A189715     m, k = n, f(n)
%o A189715     while m != k: m, k = k, f(k)
%o A189715     return m # _Chai Wah Wu_, Feb 15 2025
%Y A189715 Cf. A007913, A156595, A189716, A189717.
%Y A189715 Union of A055040 and A055047.
%K A189715 nonn
%O A189715 1,2
%A A189715 _Clark Kimberling_, Apr 26 2011
%E A189715 Name enhanced by _Peter Munn_, May 17 2020