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

1,1

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.

Links

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

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)

Example

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).

Prog

(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;
(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

Crossrefs

Cf. A014311, A057168.

Sequence in context: A351063 A070792 A178893 * A006507 A257584 A035404

Adjacent sequences: A340159 A340160 A340161 * A340163 A340164 A340165

Keyword

nonn,base

Author

Marius A. Burtea, Dec 30 2020

Status

approved