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A382745 If k appears, 7*k does not.

%I A382745 #32 Apr 11 2025 03:34:38
%S A382745 1,2,3,4,5,6,8,9,10,11,12,13,15,16,17,18,19,20,22,23,24,25,26,27,29,
%T A382745 30,31,32,33,34,36,37,38,39,40,41,43,44,45,46,47,48,49,50,51,52,53,54,
%U A382745 55,57,58,59,60,61,62,64,65,66,67,68,69,71,72,73,74,75,76,78,79,80,81,82,83,85
%N A382745 If k appears, 7*k does not.
%C A382745 Also numbers with an even number of 7's in their prime factorization.
%C A382745 Natural density 7/8.
%H A382745 Jan Snellman, <a href="/A382745/b382745.txt">Table of n, a(n) for n = 1..8751</a>
%H A382745 Jan Snellman, <a href="https://arxiv.org/abs/2504.02795">Greedy Regular Convolutions</a>, arXiv:2504.02795 [math.NT], 2025.
%F A382745 a(n) ~ (8/7)*n.
%e A382745 7 is removed since 7 = 7*1, 14, 21, 28, 35, 42 are removed, but 49 remains.
%p A382745 q:= n-> is(irem(padic[ordp](n,7), 2)=0):
%p A382745 select(q, [$1..85])[];  # _Alois P. Heinz_, Apr 04 2025
%t A382745 Select[Range[100], EvenQ[IntegerExponent[#, 7]] &] (* _Amiram Eldar_, Apr 04 2025 *)
%o A382745 (Python)
%o A382745 def ok(n):
%o A382745     c = 0
%o A382745     while n and n%7 == 0: n //= 7; c += 1
%o A382745     return c&1 == 0
%o A382745 print([k for k in range(1, 86) if ok(k)]) # _Michael S. Branicky_, Apr 04 2025
%o A382745 (Python)
%o A382745 from sympy import integer_log
%o A382745 def A382745(n):
%o A382745     def f(x): return n+x-sum((k:=x//7**m)-k//7 for m in range(0,integer_log(x,7)[0]+1,2))
%o A382745     m, k = n, f(n)
%o A382745     while m != k: m, k = k, f(k)
%o A382745     return m # _Chai Wah Wu_, Apr 10 2025
%Y A382745 Cf. A003159, A007417, A382744, A382746.
%K A382745 nonn,easy
%O A382745 1,2
%A A382745 _Jan Snellman_, Apr 04 2025