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A300630
Positive numbers k without two consecutive ones in the binary representation of 1/k.
7
1, 2, 3, 4, 6, 7, 8, 12, 14, 15, 16, 24, 28, 30, 31, 32, 48, 51, 56, 60, 62, 63, 64, 96, 102, 112, 120, 124, 126, 127, 128, 192, 195, 204, 224, 240, 248, 252, 254, 255, 256, 384, 390, 399, 408, 448, 451, 455, 480, 496, 504, 508, 510, 511, 512, 768, 771, 775
OFFSET
1,2
COMMENTS
Equivalently, these are the numbers k such that A300655(k) = 1.
Equivalently, these are the numbers k such that A300653(k, 3) > 3.
If n belongs to this sequence then 2*n belongs to this sequence.
This sequence has similarities with the Fibbinary numbers (A003714); here 1/k has no two consecutive ones in binary, there k has no two consecutive ones in binary.
For any odd term k, there is at least one positive Fibbinary number, say f, such that k * f belongs to A000225.
Apparently, the only Fibbinary numbers that belong to this sequence are the powers of 2 (A000079).
See A300669 for the complementary sequence.
Includes 2^k-1 for all k>=1. - Robert Israel, Jun 27 2018
LINKS
EXAMPLE
The first terms, alongside the binary representation of 1/a(n), are:
n a(n) bin(1/a(n)) with repeating digits in parentheses
-- ---- ------------------------------------------------
1 1 1.(0)
2 2 0.1(0)
3 3 0.(01)
4 4 0.01(0)
5 6 0.0(01)
6 7 0.(001)
7 8 0.001(0)
8 12 0.00(01)
9 14 0.0(001)
10 15 0.(0001)
11 16 0.0001(0)
12 24 0.000(01)
13 28 0.00(001)
14 30 0.0(0001)
15 31 0.(00001)
16 32 0.00001(0)
17 48 0.0000(01)
18 51 0.(00000101)
19 56 0.000(001)
20 60 0.00(0001)
MAPLE
filter:= proc(n) local m, d, r;
m:= n/2^padic:-ordp(n, 2);
d:= numtheory:-order(2, m);
r:=(2^d-1)/m;
Bits:-Or(r, 2*r)=3*r
end proc:
select(filter, [$1..1000]); # Robert Israel, Jun 27 2018
PROG
(PARI) is(n) = my (f=1/max(2, n), s=Set()); while (!setsearch(s, f), if (floor(f*4)==3, return (0), s=setunion(s, Set(f)); f=frac(f*2))); return (1)
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Rémy Sigrist, Mar 10 2018
STATUS
approved