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A328161
Numbers n that are prime or whose proper divisors (greater than 1 and less than n) have no consecutive divisibilities.
10
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
OFFSET
1,2
LINKS
EXAMPLE
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.
MAPLE
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:
select(filter, [$1..100]); # Robert Israel, Oct 11 2019
MATHEMATICA
Select[Range[100], !MatchQ[DeleteCases[Divisors[#], 1|#], {___, x_, y_, ___}/; Divisible[y, x]]&]
CROSSREFS
Equals the union of A328028 and A000040.
Complement of A328189.
One, primes, and positions of 1's in A328194.
Partitions with no consecutive divisibilities are A328171.
Sequence in context: A356438 A325456 A342524 * A003796 A032896 A032855
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 06 2019
STATUS
approved