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A162536
a(n) is the smallest positive multiple k of n such that every length of the runs of 0's and 1's in the binary representation of k divides n.
3
1, 2, 21, 4, 5, 6, 21, 16, 81, 10, 341, 12, 1365, 42, 285, 16, 85, 18, 87381, 20, 21, 22, 1398101, 24, 125, 26, 81, 84, 89478485, 90, 341, 256, 1815, 102, 1365, 36, 22906492245, 38, 117, 80, 349525, 42, 5461, 44, 4545, 598, 23456248059221, 48, 1029, 50, 1479, 52
OFFSET
1,2
COMMENTS
By "run" of 0's or 1's, it is meant: Think of binary k as a string of 0's and 1's. A single run of the digit b (0 or 1) is made up completely of consecutive digits all equal to b, and is bounded on its ends by either the digit 1-b or the end of the string.
LINKS
EXAMPLE
For n = 9, we check: 9 in binary is 1001, which has a run of two 0's, and 2 does not divide 9. Checking further: 2*9 = 18 = 10010, which still doesn't work. 3*9 = 27 = 11011 in binary, which has two runs of two 1's. 4*9 = 36 = 100100 in binary, 5*9 = 45 = 101101 in binary, 6*9 = 54 = 110110 in binary, 7*9 = 63 = 111111 in binary, 8*9 = 72 = 1001000 in binary, none of which work. But 9*9 = 81 = 1010001 in binary, which has three runs of one 1 each, a run of one 0, and a run of three 0's. Since 9 is divisible by both of these lengths (1 and 3), a(9) = 81.
MATHEMATICA
a[n_] := Block[{m}, If[n>2 && PrimeQ[n], m=1; While[Mod[m, n] > 0, m=4*m+1], m=n; While[! AllTrue[ Union[ Length /@ Split[ IntegerDigits[m, 2]]], Mod[n, #] == 0 &], m += n]]; m]; Array[a, 60] (* Giovanni Resta, Aug 11 2019 *)
CROSSREFS
Sequence in context: A303216 A331460 A303218 * A377413 A100980 A360980
KEYWORD
nonn,base
AUTHOR
Leroy Quet, Jul 05 2009
EXTENSIONS
More terms from Sean A. Irvine, Jan 26 2011
More terms from Giovanni Resta, Aug 11 2019
STATUS
approved