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

0,7

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.

Links

Kevin Ryde, Table of n, a(n) for n = 0..10000

Index entries for sequences related to binary expansion of n

Example

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

Mathematica

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

Prog

(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

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).

Sequence in context: A037882 A024865 A025109 * A048735 A102037 A286530

Adjacent sequences: A348707 A348708 A348709 * A348711 A348712 A348713

Keyword

base,easy,nonn

Author

Kevin Ryde, Oct 30 2021

Status

approved