

A161821


a(n) = the smallest integer > n 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



3, 5, 7, 9, 6, 11, 15, 17, 12, 21, 13, 19, 14, 23, 31, 33, 24, 20, 25, 37, 22, 26, 27, 35, 28, 43, 29, 39, 30, 47, 63, 65, 48, 40, 49, 73, 38, 41, 51, 69, 44, 85, 45, 50, 46, 53, 55, 67, 56, 52, 57, 75, 54, 58, 59, 71, 60, 87, 61, 79, 62, 95, 127, 129, 96, 80, 97, 72, 70, 81, 99
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,1


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 integer greater than n whose set of runlengths is a permutation of the set of runlengths for n. (See example.)


LINKS



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 integer > 84 with two runs of one 0 each and one run of two 0's (and no other runs of 0's). For example, 88 in binary is 1011000, 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 149 = 10010101 in binary. So a(84) = 149.


MAPLE

rtype := proc(n) local rt, bdgs, pr, i, rl ; rt := [seq(0, i=1..40)] ; bdgs := convert(n, base, 2) ; pr := 1 ; for i from 1 to nops(bdgs) do if op(i, bdgs) = 0 then if pr = 1 then rl := 0 ; fi; rl := rl+1 ; else if pr = 0 then rt := subsop(rl=op(rl, rt)+1, rt) ; fi; fi; pr := op(i, bdgs) ; if i = nops(bdgs) and pr = 0 then rt := subsop(rl=op(rl, rt)+1, rt) ; fi; od: rt ; end: A161821 := proc(n) local rtn, a; rtn := rtype(n) ; for a from n+1 do if rtype(a) = rtn then RETURN(a) ; fi; od: end: seq(A161821(n), n=1..100) ; # R. J. Mathar, Jul 20 2009


MATHEMATICA

f[n_] := Sort@ Map[Length, Select[Split@ IntegerDigits[n, 2], First@ # == 0 &]]; Table[Block[{k = n + 1}, While[f@ k != f@ n, k++]; k], {n, 71}] (* Michael De Vlieger, Aug 31 2017 *)


CROSSREFS



KEYWORD

base,nonn


AUTHOR



EXTENSIONS



STATUS

approved



