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A330083
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a(n) is the smallest number k > 0 such that for each b = 2..n the base-b expansion of k has exactly n - b zeros.
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0
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OFFSET
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2,2
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COMMENTS
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This list is complete. Proof: When converting base 2 to base 4, we can group the digits in base 2 into pairs from the least significant bit. We then convert pairs into single digits in base 4 as 00 -> 0, 01 -> 1, 10 -> 2, 11 -> 3. This always causes the number of zeros to go to half or less than half. For all n >= 7, n-4 is greater than (n-2)/2, so the condition is impossible. - Christopher Cormier, Dec 08 2019
Does k exist for every n >= 2?
a(7) > 10^7, if it exists.
a(7) > 2^64, if it exists. - Giovanni Resta, Dec 01 2019
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LINKS
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Table of n, a(n) for n=2..6.
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EXAMPLE
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For n = 6: The base-b expansions of 271 for b = 2..6 are shown in the following table:
b | base-b expansion | number of zeros
---------------------------------------
2 | 100001111 | 4
3 | 101001 | 3
4 | 10033 | 2
5 | 2041 | 1
6 | 1131 | 0
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PROG
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(PARI) count_zeros(vec) = #setintersect(vecsort(vec), vector(#vec))
a(n) = for(k=1, oo, for(b=2, n, if(count_zeros(digits(k, b))!=n-b, break, if(b==n, return(k)))))
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CROSSREFS
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Sequence in context: A254059 A346551 A180591 * A322951 A156446 A032685
Adjacent sequences: A330080 A330081 A330082 * A330084 A330085 A330086
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KEYWORD
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nonn,base,fini,full
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AUTHOR
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Felix Fröhlich, Dec 01 2019
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EXTENSIONS
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Value of a(2) adjusted by Felix Fröhlich, Dec 09 2019
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STATUS
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approved
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