A348710 In the binary expansion of n, decrease the length of each run of 1-bits by one.

0, 0, 0, 1, 0, 0, 2, 3, 0, 0, 0, 1, 4, 2, 6, 7, 0, 0, 0, 1, 0, 0, 2, 3, 8, 4, 4, 5, 12, 6, 14, 15, 0, 0, 0, 1, 0, 0, 2, 3, 0, 0, 0, 1, 4, 2, 6, 7, 16, 8, 8, 9, 8, 4, 10, 11, 24, 12, 12, 13, 28, 14, 30, 31, 0, 0, 0, 1, 0, 0, 2, 3, 0, 0, 0, 1, 4, 2, 6, 7, 0, 0, 0

Offset: 0

Author

Kevin Ryde, Oct 30 2021

Keywords

• base
• easy
• nonn

Comments

Equivalently, change bits 01 -> 0, including a 0 reckoned above the most significant 1-bit of n so change there.

A single 1-bit run decreases to nothing. The Fibbinary numbers (A003714) are those n with only single 1-bits so that a(n) = 0 iff n is in A003714.

a(n) = 1 iff n is in A213540 since those values end with bits 011 (which become 01) and otherwise have only single 1-bits, as do the Fibbinary numbers.

Decreasing each run is the inverse of the increase A175048 so that a(A175048(k)) = k. This n = A175048(k) is the smallest n with a(n) = k and then other occurrences of k are by inserting single 1-bits into this n, including anywhere above the most significant bit.

Examples

			n    = 14551 = binary 111 000 11 0 1 0 111
a(n) =   787 = binary  11 000  1 0   0  11
		

Links

• Kevin Ryde, Table of n, a(n) for n = 0..10000
• Index entries for sequences related to binary expansion of n

Crossrefs

Cf. A007088 (binary), A175048 (increase 1-bits), A090077 (decrease to single 1-bits).

Cf. A003714 (indices of 0's), A213540 (indices of 1's).

Cf. A106151 (decrease 0-bits), A318921 (decrease each run).

Programs

Mathematica

Table[FromDigits[Flatten[Split@IntegerDigits[n,2]/. {1,a___}:>{a}],2],{n,0,82}] (* Giorgos Kalogeropoulos, Nov 01 2021 *)

PARI

a(n) = my(v=binary(n),t=0); for(i=2,#v, if(v[i-1]||!v[i], v[t++]=v[i])); fromdigits(v[1..t],2);

Python

def a(n): return int(bin(n).replace("b", "").replace("01", "0"), 2)
print([a(n) for n in range(83)]) # Michael S. Branicky, Oct 31 2021