

A161822


a(n) = the smallest positive integer such that both n and a(n), when represented in binary, contain the same types of runs of 0's, the runs being in any order.


5



1, 2, 1, 4, 2, 2, 1, 8, 4, 10, 2, 4, 2, 2, 1, 16, 8, 18, 4, 18, 10, 10, 2, 8, 4, 10, 2, 4, 2, 2, 1, 32, 16, 34, 8, 36, 18, 18, 4, 34, 18, 42, 10, 18, 10, 10, 2, 16, 8, 18, 4, 18, 10, 10, 2, 8, 4, 10, 2, 4, 2, 2, 1, 64, 32, 66, 16, 68, 34, 34, 8, 68, 36, 74, 18, 36, 18, 18, 4, 66, 34, 74, 18
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OFFSET

1,2


COMMENTS

Clarification of definition: Think of binary n and a(n) each as a string of 0's and 1's. Consider the "runs" of 0's in binary n and a(n), where each run is made up completely of 0's, and is bounded on both sides either by 1's or by the edge of the string. Now consider the lengths of each bounded run of 0's (the number of 0's in each run). Then a(n) is the smallest positive integer whose set of runlengths is a permutation of the set of runlengths for n. (See example.)
1 <= a(n) <= n. a(n) = n iff n = 2^m for some m. a(n) = 1 iff n = 2^m  1 for some m.  Hagen von Eitzen, Jun 22 2009


LINKS

Michael De Vlieger, Table of n, a(n) for n = 1..10000


EXAMPLE

84 in binary is 1010100. There are three runs of 0's, two runs of one 0 each and one run of two 0's. So we are looking for the smallest positive integer with two runs of one 0 each and one run of two 0's (and no other runs of 0's). For example, 16 in binary is 10000, which contains the runs, except that it is required that each run be bounded by 1's or the edge of the binary string. The next number that fits the requirements completely is 74 = 1001010 in binary. So a(84) = 74.


MATHEMATICA

f[n_] := Sort@ Map[Length, Select[Split@ IntegerDigits[n, 2], First@ # == 0 &]]; Table[Block[{k = 1}, While[f@ k != f@ n, k++]; k], {n, 83}] (* Michael De Vlieger, Aug 31 2017 *)


PROG

(PARI) a(n)=local(B=binary(n), L=[], ct=0); forstep(i=#B, 1, 1, if(B[i], if(ct, L=concat(L, ct); ct=0), ct++)); L=vecsort(L); forstep(i=#L, 1, 1, ct=(ct*2+1)*2^L[i]); if(ct>0, ct, 1) \\ Hagen von Eitzen, Jun 22 2009


CROSSREFS

Cf. A161819, A161820, A161821.
Sequence in context: A109090 A220780 A080100 * A001176 A136693 A086685
Adjacent sequences: A161819 A161820 A161821 * A161823 A161824 A161825


KEYWORD

base,nonn


AUTHOR

Leroy Quet, Jun 20 2009


EXTENSIONS

More terms from Hagen von Eitzen, Jun 22 2009


STATUS

approved



