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

Author

Vladimir Shevelev, Dec 13 2013

Keywords

• nonn
• base

Comments

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:

0: () 38: (3,1,2) 98: (1,4,2)

1: (1) 40: (2,4) 104: (1,2,4)

2: (2) 41: (2,3,1) 128: (8)

4: (3) 44: (2,1,3) 129: (7,1)

5: (2,1) 48: (1,5) 130: (6,2)

6: (1,2) 50: (1,3,2) 132: (5,3)

8: (4) 52: (1,2,3) 133: (5,2,1)

9: (3,1) 64: (7) 134: (5,1,2)

12: (1,3) 65: (6,1) 137: (4,3,1)

16: (5) 66: (5,2) 140: (4,1,3)

17: (4,1) 68: (4,3) 144: (3,5)

18: (3,2) 69: (4,2,1) 145: (3,4,1)

20: (2,3) 70: (4,1,2) 152: (3,1,4)

24: (1,4) 72: (3,4) 160: (2,6)

32: (6) 80: (2,5) 161: (2,5,1)

33: (5,1) 81: (2,4,1) 176: (2,1,5)

34: (4,2) 88: (2,1,4) 192: (1,7)

37: (3,2,1) 96: (1,6) 194: (1,5,2)

(End)

Examples

			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.
		

Links

• Index entries for sequences related to binary expansion of n

Crossrefs

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.

Programs

Mathematica

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 *)

Extensions

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

0 prepended by Gus Wiseman, Apr 04 2020