A233655 -id:A233655 - cp's OEIS frontend

A233564 c-squarefree numbers: positive integers which in binary are concatenation of distinct parts of the form 10...0 with nonnegative number of zeros.

0, 1, 2, 4, 5, 6, 8, 9, 12, 16, 17, 18, 20, 24, 32, 33, 34, 37, 38, 40, 41, 44, 48, 50, 52, 64, 65, 66, 68, 69, 70, 72, 80, 81, 88, 96, 98, 104, 128, 129, 130, 132, 133, 134, 137, 140, 144, 145, 152, 160, 161, 176, 192, 194, 196, 200, 208, 256, 257, 258, 260, 261

Offset: 1

Vladimir Shevelev, Dec 13 2013

Number of terms in interval [2^(n-1), 2^n) is the number of compositions of n with distinct parts (cf. A032020). For example, if n=6, then interval [2^5, 2^6) contains  11 terms {32,...,52}. This corresponds to 11 compositions with distinct parts of 6: 6, 5+1, 1+5, 4+2, 2+4, 3+2+1, 3+1+2, 2+3+1, 2+1+3, 1+3+2, 1+2+3.
From Gus Wiseman, Apr 06 2020: (Start)
The k-th composition in standard order (row k of A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions. This sequence lists all numbers k such that the k-th composition in standard order is strict. For example, the sequence together with the corresponding strict compositions begins:
()          38: (3,1,2)     98: (1,4,2)
(1)         40: (2,4)      104: (1,2,4)
(2)         41: (2,3,1)    128: (8)
(3)         44: (2,1,3)    129: (7,1)
(2,1)       48: (1,5)      130: (6,2)
(1,2)       50: (1,3,2)    132: (5,3)
(4)         52: (1,2,3)    133: (5,2,1)
(3,1)       64: (7)        134: (5,1,2)
(1,3)       65: (6,1)      137: (4,3,1)
(5)         66: (5,2)      140: (4,1,3)
(4,1)       68: (4,3)      144: (3,5)
(3,2)       69: (4,2,1)    145: (3,4,1)
(2,3)       70: (4,1,2)    152: (3,1,4)
(1,4)       72: (3,4)      160: (2,6)
(6)         80: (2,5)      161: (2,5,1)
(5,1)       81: (2,4,1)    176: (2,1,5)
(4,2)       88: (2,1,4)    192: (1,7)
(3,2,1)     96: (1,6)      194: (1,5,2)
(End)

			49 in binary has the following parts of the form 10...0 with nonnegative number of  zeros: (1),(1000),(1). Two of them are the same. So it is not in the sequence. On the other hand, 50 has distinct parts (1)(100)(10), thus it is a term.

Index entries for sequences related to binary expansion of n

Cf. A032020, A124771, A233249, A233312, A233416, A233420, A233569, A233655.
A subset of A333489 and superset of A333218.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Weighted sum is A029931.
- Partial sums from the right are A048793.
- Sum is A070939.
- Runs are counted by A124767.
- Reversed initial intervals A164894.
- Initial intervals are A246534.
- Constant compositions are A272919.
- Strictly decreasing compositions are A333255.
- Strictly increasing compositions are A333256.
- Anti-runs are counted by A333381.
- Anti-runs are A333489.
Cf. A114994, 225620, A228351, A238279, A242882, A329739, A329744, A333217.

bitPatt[n_]:=bitPatt[n]=Split[IntegerDigits[n,2],#1>#2||#2==0&];
Select[Range[0,300],bitPatt[#]==DeleteDuplicates[bitPatt[#]]&] (* Peter J. C. Moses, Dec 13 2013 *)
stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],UnsameQ@@stc[#]&] (* Gus Wiseman, Apr 04 2020 *)

More terms from Peter J. C. Moses, Dec 13 2013

0 prepended by Gus Wiseman, Apr 04 2020

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.

Original entry on oeis.org

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

Leroy Quet, Jul 03 2009

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)

			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.

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

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

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

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

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

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

A235669 Sum of parts of the form 10...0 and 20...0 with nonnegative number of zeros in ternary representation of n as the corresponding numbers 3^n and 2*3^n.

Original entry on oeis.org

0, 1, 2, 3, 2, 3, 6, 3, 4, 9, 4, 5, 4, 3, 4, 7, 4, 5, 18, 7, 8, 5, 4, 5, 8, 5, 6, 27, 10, 11, 6, 5, 6, 9, 6, 7, 10, 5, 6, 5, 4, 5, 8, 5, 6, 19, 8, 9, 6, 5, 6, 9, 6, 7, 54, 19, 20, 9, 8, 9, 12, 9, 10, 11, 6, 7, 6, 5, 6, 9, 6, 7, 20, 9, 10, 7, 6, 7, 10, 7, 8, 81, 28, 29, 12

Offset: 0

Vladimir Shevelev, Jan 13 2014

The number of appearances of k is the number of compositions of k into numbers of the form 3^n and 2*3^n, A235684(k).

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A124771, A162439, A233249, A233312, A233416, A233420, A233564, A233569, A233655.

bitPatt[n_,b_]:=Split[IntegerDigits[n,b ],#2==0&]; Map[Plus@@Map[FromDigits[#,3]&,bitPatt[#,3]]&,Range[0,50]] (* Peter J. C. Moses, Jan 13 2014 *)

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A233564 c-squarefree numbers: positive integers which in binary are concatenation of distinct parts of the form 10...0 with nonnegative number of zeros.

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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.

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A235669 Sum of parts of the form 10...0 and 20...0 with nonnegative number of zeros in ternary representation of n as the corresponding numbers 3^n and 2*3^n.

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Mathematica