A093049 n-1 minus exponent of 2 in n, a(0) = 0.

0, 0, 0, 2, 1, 4, 4, 6, 4, 8, 8, 10, 9, 12, 12, 14, 11, 16, 16, 18, 17, 20, 20, 22, 20, 24, 24, 26, 25, 28, 28, 30, 26, 32, 32, 34, 33, 36, 36, 38, 36, 40, 40, 42, 41, 44, 44, 46, 43, 48, 48, 50, 49, 52, 52, 54, 52, 56, 56, 58, 57, 60, 60, 62, 57, 64, 64, 66, 65, 68

Offset: 0

Author

Ralf Stephan, Mar 16 2004

Keywords

• nonn

Examples

			G.f. = 2*x^3 + x^4 + 4*x^5 + 4*x^6 + 6*x^7 + 4*x^8 + 8*x^9 + 8*x^10 + ... - _Michael Somos_, Jan 25 2020
		

Crossrefs

a(n) = n - A007814(n) - 1 = A093048(n) - 1, n>0.

a(n) is the exponent of 2 in A001761(n+1), A002105(n), A002682(n-1), A006963(n), A036770(n-1), A059837(n), A084623(n), |A003707(n)|, |A011859(n)|.

Programs

Mathematica

a[ n_] := If[ n == 0, 0, n - 1 - IntegerExponent[n, 2]]; (* Michael Somos, Jan 25 2020 *)

PARI

a(n)=if(n<1,0,if(n%2==0,a(n/2)+n/2-1,n-1))

PARI

{a(n) = if( n, n - 1 - valuation(n, 2))}; /* Michael Somos, Jan 25 2020 */

Python

def A093049(n): return n-1-(~n& n-1).bit_length() if n else 0 # Chai Wah Wu, Jul 07 2022

Formula

Recurrence: a(2n) = a(n) + n - 1, a(2n+1) = 2n.

G.f.: sum(k>=0, t^3(t+2)/(1-t^2)^2, t=x^2^k).