A162439 Write down the binary representation of n. Partition the string which is this binary representation by placing a '+' just left of every 1. Add the resulting base 2 numbers. a(n) = decimal equivalent of this sum.

1, 2, 2, 4, 3, 3, 3, 8, 5, 4, 4, 5, 4, 4, 4, 16, 9, 6, 6, 6, 5, 5, 5, 9, 6, 5, 5, 6, 5, 5, 5, 32, 17, 10, 10, 8, 7, 7, 7, 10, 7, 6, 6, 7, 6, 6, 6, 17, 10, 7, 7, 7, 6, 6, 6, 10, 7, 6, 6, 7, 6, 6, 6, 64, 33, 18, 18, 12, 11, 11, 11, 12, 9, 8, 8, 9, 8, 8, 8, 18, 11, 8, 8, 8, 7, 7, 7, 11, 8, 7, 7, 8, 7, 7

Offset: 1

Author

Leroy Quet, Jul 03 2009

Keywords

• base
• nonn
• look

Comments

From Vladimir Shevelev, Dec 11 2014: (Start)

Or, sum of parts of the form 10...0 with nonnegative number of zeros in binary representation of n as the corresponding powers of 2. For example, n=50 in binary is a concatenation of parts (1)(100)(10). Then a(50)=1+4+2=7.

Every positive number k occurs a finite number of times, such that the position of the last appearance of k is 2^k-1.

Moreover, the number of times of appearances of k is the number of compositions of k into powers of 2, i.e., it is A023359(k), k>0. (End)

Examples

			52 in binary is 110100. Placing the +'s before every 1, we get +1+10+100, which is 1+2+4 = 7 in decimal. So a(52) = 7.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A124771, A233249, A233312, A233416, A233420, A233564, A233569, A233655.

Programs

Maple

a:= proc(n) local l, s, i, j; l:= convert(n, base, 2); s:= 0; i:=1; for j from nops(l)-1 to 1 by -1 do if l[j]=0 then i:= i*2; else s:= s+i; i:= 1 fi od; s+i end: seq(a(n), n=1..150); # Alois P. Heinz, Jul 28 2009
Lton := proc(L) local i ; add(op(i,L)*2^(i-1),i=1..nops(L)) ; end: A162439 := proc(n) local a,lef,b2,ri ; a := 0 ; lef := 0; b2 := convert(n,base,2) ; for ri from lef+1 do if op(ri,b2) = 1 then a := a+Lton([op(lef+1..ri,b2)]) ; lef := ri ; fi; if ri =nops(b2) then break; fi; od: a ; end: seq(A162439(n),n=1..100) ; # R. J. Mathar, Jul 30 2009

Mathematica

a[n_] := FromDigits[#, 2]& /@ Split[IntegerDigits[n, 2] , #2==0&] // Total; Array[a, 100] (* Jean-François Alcover, Jan 07 2016 *)

Formula

Let, for k_1>k_2>...>k_r, n = 2^k_1 + 2^k_2 +...+ 2^k_r. Then a(n) = 2^(k_1-k_2-1) + 2^(k_2-k_3-1) + 2^(k_(r-1)-k_r-1) + 2^k_r. - Vladimir Shevelev, Dec 11 2013

Extensions

More terms from Alois P. Heinz and R. J. Mathar, Jul 28 2009