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A316534
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Numbers k such that k concatenated with k+1 and then divided by 2k+1 produces an integer after a series of divisions explained in the Example section.
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3
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1, 2, 3, 4, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 59, 73, 74, 75, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119
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OFFSET
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1,2
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LINKS
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EXAMPLE
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1 is in the sequence because 12/(1+2) is the integer 4;
2 is in the sequence though 23/(2+3) is not an integer because if we compute floor(23/(2+3)) we get 4, then if we use this 4 to compute floor(34/(3+4)) we get 4, then again floor(44/(4+4)) = 5 and in the end 45/(4+5) is the integer 5;
3 is in the sequence though 34/(3+4) is not an integer, because if we apply the "floor" trick again, we will end in an integer: floor(34/(3+4)) = 4, then floor(44/(4+4)) = 5 and 45/(4+5) is the integer 5;
4 is in the sequence because 45/(4+5) is the integer 5;
5 is not in the sequence because 56/(5+6) is not an integer and even if we repeatedly apply the "floor" trick, we will be stuck in a loop: floor(56/(5+6)) = 5, then floor(65/(6+5)) = 5, then floor(55/(5+5)) = 5, then again floor(55/(5+5)) = 5, etc. So 5 will never produce an integer at the end.
6 is not in the sequence for the same reason: floor(67/(6+7)) = 5, then floor(75/(7+5)) = 6, then floor(56/(5+6)) = 5, then floor(65/(6+5)) = 5, then floor(55/(5+5)) = 5, then again floor(55/(5+5)) = 5, etc. So 6 will never produce an integer at the end.
7 is not in the sequence for the same reason again: floor(78/(7+8)) = 5, then floor(85/(8+5)) = 6, then floor(56/(5+6)) = 5, then floor(65/(6+5)) = 5, then floor(55/(5+5)) = 5, then again floor(55/(5+5)) = 5, etc. So 7 will never produce an integer at the end.
. . .
10 is in the sequence though 1011/(10+11) is not an integer, because if we apply the "floor" trick again, we will end on an integer: floor(1011/(10+11)) = 48, then floor(1148/(11+48)) = 19, then floor(4819/(48+19)) = 71, then floor(1971/(19+71)) = 21, then floor(7121/(71+21)) = 77, then floor(2177/(21+77) = 22, then 7722/(77+22) is the integer 78.
Etc.
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CROSSREFS
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Cf. A316538 where the concatenation k and k-1 is considered (instead of k and k+1 here).
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KEYWORD
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base,nonn
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AUTHOR
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STATUS
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approved
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