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A162536
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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.
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3
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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
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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