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).

7, 14, 19, 28, 37, 56, 38, 112, 74, 224, 76, 448, 148, 896, 152, 1792, 296, 3584, 304, 7168, 592, 14336, 608, 28672, 1184, 57344, 1216, 114688, 2368, 229376, 2432, 458752, 4736, 917504, 4864, 1835008, 9472, 3670016, 9728, 7340032, 18944, 14680064, 19456, 29360128

Offset: 1

Author

Marius A. Burtea, Dec 30 2020

Keywords

• nonn
• base

Comments

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.

Examples

			a(1) * a(2) = 7 * 14 = A014311(1) * A014311(4) = A014311(32).
a(2) * a(3) = 14 * 19 = A014311(4) * A014311(5) = A014311(61).
a(3) * a(4) = 19 * 28 = A014311(5) * A014311(10) = A014311(93).
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A014311, A057168.

Programs

Magma

fb:=func; 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;

PARI

isokd(n) = hammingweight(n) == 3; \\ A014311
nexth(n) = my(u=bitand(n, -n), v=u+n); (bitxor(v, n)/u)>>2+v; \\ A057168
nextk(va, n) = {my(ok = 0, k = 7); while (! (isokd(k*va[n-1]) && !#select(x->(x==k), va)), k = nexth(k)); k;}
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

Formula

Conjectures from Chai Wah Wu, Jan 27 2021: (Start)

a(n) = 2*a(n-2) + 2*a(n-4) - 4*a(n-6) for n > 7.

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)