A115845 Numbers n such that there is no bit position where the binary expansions of n and 8n are both 1.

0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 16, 17, 20, 21, 24, 28, 32, 33, 34, 35, 40, 42, 48, 49, 56, 64, 65, 66, 67, 68, 69, 70, 71, 80, 81, 84, 85, 96, 97, 98, 99, 112, 113, 128, 129, 130, 131, 132, 133, 134, 135, 136, 138, 140, 142, 160, 161, 162, 163, 168, 170, 192

Offset

1,3

Comments

Equivalently, numbers n such that 9*n = 9 X n, i.e., 8*n XOR n = 9*n. Here * stands for ordinary multiplication and X means carryless (GF(2)[X]) multiplication (A048720).

Equivalently, numbers n such that the binomial coefficient C(9n,n) (A169958) is odd. - Zak Seidov, Aug 06 2010

The equivalence of these three definitions follows from Lucas's theorem on binomial coefficients. - N. J. A. Sloane, Sep 01 2010

Clearly all numbers k*2^i for 1 <= k <= 7 have this property. - N. J. A. Sloane, Sep 01 2010

A116361(a(n)) <= 3. - Reinhard Zumkeller, Feb 04 2006

Links

N. J. A. Sloane and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Index entries for sequences defined by congruent products between domains N and GF(2)[X]

Index entries for sequences defined by congruent products under XOR

Formula

a(n)/n^k is bounded (but does not tend to a limit), where k = 1.44... = A104287. - Charles R Greathouse IV, Sep 23 2012

Mathematica

Reap[Do[If[OddQ[Binomial[9n, n]], Sow[n]], {n, 0, 400}]][[2, 1]] (* Zak Seidov, Aug 06 2010 *)

Prog

(PARI) is(n)=!bitand(n, n<<3) \\ Charles R Greathouse IV, Sep 23 2012

Crossrefs

A115846 shows this sequence in binary.

A033052 is a subsequence.

Cf. A003714, A048716, A115847, A116360, A005809, A003714, A048716, A048715.

Sequence in context: A178878 A175326 A018676 * A026507 A079645 A316114

Adjacent sequences: A115842 A115843 A115844 * A115846 A115847 A115848

Keyword

nonn

Author

Antti Karttunen, Feb 01 2006

Extensions

Edited with a new definition by N. J. A. Sloane, Sep 01 2010, merging this sequence with a sequence submitted by Zak Seidov, Aug 06 2010

Status

approved