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A340162 a(1) = 7; thereafter a(n) is the smallest number k with exactly three 1-bits, not already in the sequence, for which k*a(n - 1) has exactly three 1-bits (A014311).

%I A340162 #23 Sep 08 2022 08:46:26
%S A340162 7,14,19,28,37,56,38,112,74,224,76,448,148,896,152,1792,296,3584,304,
%T A340162 7168,592,14336,608,28672,1184,57344,1216,114688,2368,229376,2432,
%U A340162 458752,4736,917504,4864,1835008,9472,3670016,9728,7340032,18944,14680064,19456,29360128
%N A340162 a(1) = 7; thereafter a(n) is the smallest number k with exactly three 1-bits, not already in the sequence, for which k*a(n - 1) has exactly three 1-bits (A014311).
%C A340162 It seems that a(2*k) = 2^k*7, a(4*k - 1) = 2^(k - 1)*19, a(4*k + 1) = 2^(k - 1)*37, k >= 1.
%H A340162 Chai Wah Wu, <a href="/A340162/b340162.txt">Table of n, a(n) for n = 1..1000</a>
%F A340162 Conjectures from _Chai Wah Wu_, Jan 27 2021: (Start)
%F A340162 a(n) = 2*a(n-2) + 2*a(n-4) - 4*a(n-6) for n > 7.
%F A340162 G.f.: x*(-46*x^6 - 28*x^5 - 15*x^4 + 5*x^2 + 14*x + 7)/((2*x^2 - 1)*(2*x^4 - 1)). (End)
%e A340162 a(1) * a(2) = 7 * 14 = A014311(1) * A014311(4) = A014311(32).
%e A340162 a(2) * a(3) = 14 * 19 = A014311(4) * A014311(5) = A014311(61).
%e A340162 a(3) * a(4) = 19 * 28 = A014311(5) * A014311(10) = A014311(93).
%o A340162 (Magma) fb:=func<n|(Multiplicity(Intseq(n,2),1)) eq 3>; a:=[7]; for n in [2..44] do k:=7; while k in a or (not fb(k) or not fb(a[n-1]*k)) do k:=k+1; end while; Append(~a,k); end for; a;
%o A340162 (PARI) isokd(n) = hammingweight(n) == 3; \\ A014311
%o A340162 nexth(n) = my(u=bitand(n, -n), v=u+n); (bitxor(v, n)/u)>>2+v; \\ A057168
%o A340162 nextk(va, n) = {my(ok = 0, k = 7); while (! (isokd(k*va[n-1]) && !#select(x->(x==k), va)), k = nexth(k)); k;}
%o A340162 lista(nn) = {my(va = vector(nn)); va[1] = 7; for (n=2, nn, my(k = nextk(va, n)); va[n] = k;); va; } \\ _Michel Marcus_, Jan 14 2021
%Y A340162 Cf. A014311, A057168.
%K A340162 nonn,base
%O A340162 1,1
%A A340162 _Marius A. Burtea_, Dec 30 2020