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A325123
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Number of divisible pairs of positive integers up to n with no binary carries.
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7
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0, 0, 1, 1, 3, 3, 4, 4, 7, 7, 9, 9, 12, 12, 13, 13, 17, 17, 19, 19, 22, 22, 23, 23, 28, 28, 29, 29, 31, 31, 32, 32, 37, 37, 39, 39, 44, 44, 45, 45, 50, 50, 52, 52, 54, 54, 55, 55, 62, 62, 64, 64, 66, 66, 68, 68, 72, 72, 73, 73, 76, 76, 77, 77, 83, 83, 85, 85
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OFFSET
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0,5
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COMMENTS
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Two positive integers are divisible if the first divides the second, and they have a binary carry if the positions of 1's in their reversed binary expansion overlap.
a(2k+1) = a(2k), since an odd number and any divisor will overlap in the last digit. Additionally, a(2k+2) > a(2k+1) because the pair {1,2k+2} is always valid. Therefore, every term appears exactly twice. - Charlie Neder, Apr 02 2019
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LINKS
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EXAMPLE
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The a(2) = 1 through a(11) = 9 pairs:
{1,2} {1,2} {1,2} {1,2} {1,2} {1,2} {1,2} {1,2} {1,2} {1,2}
{1,4} {1,4} {1,4} {1,4} {1,4} {1,4} {1,4} {1,4}
{2,4} {2,4} {1,6} {1,6} {1,6} {1,6} {1,6} {1,6}
{2,4} {2,4} {1,8} {1,8} {1,8} {1,8}
{2,4} {2,4} {2,4} {2,4}
{2,8} {2,8} {2,8} {2,8}
{4,8} {4,8} {4,8} {4,8}
{1,10} {1,10}
{5,10} {5,10}
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MATHEMATICA
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Table[Length[Select[Tuples[Range[n], 2], Divisible@@Reverse[#]&&Intersection[Position[Reverse[IntegerDigits[#[[1]], 2]], 1], Position[Reverse[IntegerDigits[#[[2]], 2]], 1]]=={}&]], {n, 0, 20}]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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