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A328161
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Numbers n that are prime or whose proper divisors (greater than 1 and less than n) have no consecutive divisibilities.
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10
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1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 19, 21, 22, 23, 24, 25, 26, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 43, 45, 46, 47, 48, 49, 51, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 67, 69, 70, 71, 72, 73, 74, 77, 79, 82, 83, 84, 85, 86, 87, 89, 90, 91
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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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The proper divisors of 18 are {2, 3, 6, 9}, and {3, 6} are a consecutive divisible pair, so 18 does not belong to the sequence.
The proper divisors of 60 are {2, 3, 4, 5, 6, 10, 12, 15, 20, 30}, and none of {2, 3}, {3, 4}, {4, 5}, {5, 6}, {6, 10}, {10, 12}, {12, 15}, {15, 20}, or {20, 30} are divisible pairs, so 60 belongs to the sequence.
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MAPLE
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filter:= proc(n) local D, i;
if isprime(n) then return true fi;
D:= sort(convert(numtheory:-divisors(n) minus {1, n}, list));
for i from 1 to nops(D)-1 do if (D[i+1]/D[i])::integer then return false fi od:
true
end proc:
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MATHEMATICA
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Select[Range[100], !MatchQ[DeleteCases[Divisors[#], 1|#], {___, x_, y_, ___}/; Divisible[y, x]]&]
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CROSSREFS
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One, primes, and positions of 1's in A328194.
Partitions with no consecutive divisibilities are A328171.
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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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