A181155 Odious numbers (A000069) plus one; complement of A026147.

2, 3, 5, 8, 9, 12, 14, 15, 17, 20, 22, 23, 26, 27, 29, 32, 33, 36, 38, 39, 42, 43, 45, 48, 50, 51, 53, 56, 57, 60, 62, 63, 65, 68, 70, 71, 74, 75, 77, 80, 82, 83, 85, 88, 89, 92, 94, 95, 98, 99, 101, 104, 105, 108, 110, 111, 113, 116, 118, 119, 122, 123, 125, 128, 129, 132

Offset: 1

Author

Matthew Vandermast, Oct 06 2010

Keywords

• easy
• nonn

Comments

a(n) = position of n-th 2 in A001285 if offset for A001285 is given as 1.

It appears that this sequence and A026147 index each other's even terms (i.e., a(n) = position of n-th even term in A026147, and A026147(n) = position of n-th even term in this sequence). It also appears that each of the two sequences indexes its own odd terms (cf. A079000).

Barbeau notes that if let A = the first 2^k terms of A026147 and B = the first 2^k terms of this sequence, then the two sets have the same sum of powers for first up to the k-th power. I note it holds for 0th power also. - Michael Somos, Jun 09 2013

Examples

			Let k=2. Then A = {1,4,6,7} and B = {2,3,5,8} have the property that 1^0+4^0+6^0+7^0 = 2^0+3^0+5^0+8^0 = 4, 1^1+4^1+6^1+7^1 = 2^1+3^1+5^1+8^1 = 18, and 1^2+4^2+6^2+7^2 = 2^2+3^2+5^2+8^2 = 102. - _Michael Somos_, Jun 09 2013
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Edward J. Barbeau, Power Play, MAA, 1997. See p. 104.

Crossrefs

Cf. A026147.

Programs

Mathematica

a[ n_] := If[ n < 1, 0, 2 n - Mod[ Total[ IntegerDigits[ n - 1, 2]], 2]] (* Michael Somos, Jun 09 2013 *)

PARI

a(n)=2*n-hammingweight(n-1)%2 \\ Charles R Greathouse IV, Mar 22 2013

PARI

{a(n) = if( n<1, 0, 2*n - subst( Pol( binary( n-1)), x, 1)%2)} /* Michael Somos, Jun 09 2013 */

Python

def A181155(n): return 1+((m:=n-1)<<1)+(m.bit_count()&1^1) # Chai Wah Wu, Mar 03 2023

Formula

a(n) = A000069(n) + 1.

a(a(n)-1) = 2*a(n)-1. - Benoit Cloitre, Oct 07 2010

a(n) + A010060(n+1) = 2n + 2 for n >= 0. - Clark Kimberling, Oct 06 2014