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A161820 a(n) is the smallest positive integer such that both n and a(n), when represented in binary, contain the same types of runs of 1's, the runs being in any order. 5
1, 1, 3, 1, 5, 3, 7, 1, 5, 5, 11, 3, 11, 7, 15, 1, 5, 5, 11, 5, 21, 11, 23, 3, 11, 11, 27, 7, 23, 15, 31, 1, 5, 5, 11, 5, 21, 11, 23, 5, 21, 21, 43, 11, 43, 23, 47, 3, 11, 11, 27, 11, 43, 27, 55, 7, 23, 23, 55, 15, 47, 31, 63, 1, 5, 5, 11, 5, 21, 11, 23, 5, 21, 21, 43, 11, 43, 23, 47, 5 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
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 1's in binary n and a(n), where each run is made up completely of 1's, and is bounded on both sides either by 0's or by the edge of the string. Now consider the lengths of each bounded run of 1's (the number of 1's in each run). Then a(n) is the smallest positive integer whose set of run-lengths is a permutation of the set of run-lengths for n. (See example.)
LINKS
EXAMPLE
77 in binary is 1001101. There are three runs of 1's, two runs of one 1 each and one run of two 1's. So we are looking for the smallest positive integer with two runs of one 1 each and one run of two 1's (and no other runs of 1's). For example, 15 in binary is 1111, which contains the runs, except that it is required that each run be bounded by 0's or the edge of the binary string. The next number that fits the requirements completely is 43 = 101011 in binary. So a(77) = 43.
MAPLE
rtype := proc(n) local rt, bdgs, pr, i, rl ; rt := [seq(0, i=1..40)] ; bdgs := convert(n, base, 2) ; pr := 0 ; for i from 1 to nops(bdgs) do if op(i, bdgs) = 1 then if pr = 0 then rl := 0 ; fi; rl := rl+1 ; else if pr = 1 then rt := subsop(rl=op(rl, rt)+1, rt) ; fi; fi; pr := op(i, bdgs) ; if i = nops(bdgs) and pr = 1 then rt := subsop(rl=op(rl, rt)+1, rt) ; fi; od: rt ; end: A161820 := proc(n) local rtn, a; rtn := rtype(n) ; for a from 1 do if rtype(a) = rtn then RETURN(a) ; fi; od: end: seq(A161820(n), n=1..100) ; # R. J. Mathar, Jul 20 2009
MATHEMATICA
f[n_] := Sort@ Map[Length, Select[Split@ IntegerDigits[n, 2], First@ # == 1 &]]; Table[Block[{k = 1}, While[f@ k != f@ n, k++]; k], {n, 69}] (* Michael De Vlieger, Aug 30 2017 *)
CROSSREFS
Sequence in context: A161398 A204455 A318653 * A341042 A116528 A357111
KEYWORD
base,nonn
AUTHOR
Leroy Quet, Jun 20 2009
EXTENSIONS
More terms from R. J. Mathar, Jul 20 2009
STATUS
approved

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Last modified June 17 06:37 EDT 2024. Contains 373432 sequences. (Running on oeis4.)