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

Author

Antti Karttunen, Feb 01 2006

Keywords

• nonn

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

Crossrefs

A115846 shows this sequence in binary.

A033052 is a subsequence.

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

Programs

Mathematica

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

PARI

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

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

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