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A328028 Nonprime numbers n whose proper divisors (greater than 1 and less than n) have no consecutive divisibilities.

%I A328028 #14 Oct 30 2019 00:55:25
%S A328028 1,4,6,9,10,12,14,15,21,22,24,25,26,30,33,34,35,36,38,39,45,46,48,49,
%T A328028 51,55,57,58,60,62,63,65,69,70,72,74,77,82,84,85,86,87,90,91,93,94,95,
%U A328028 96,105,106,108,111,115,118,119,120,121,122,123,129,132,133,134
%N A328028 Nonprime numbers n whose proper divisors (greater than 1 and less than n) have no consecutive divisibilities.
%H A328028 Robert Israel, <a href="/A328028/b328028.txt">Table of n, a(n) for n = 1..10000</a>
%e A328028 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.
%e A328028 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.
%p A328028 filter:= proc(n) local D,i;
%p A328028   if isprime(n) then return false fi;
%p A328028   D:= sort(convert(numtheory:-divisors(n) minus {1,n}, list));
%p A328028   for i from 1 to nops(D)-1 do if (D[i+1]/D[i])::integer then return false fi od:
%p A328028   true
%p A328028 end proc:
%p A328028 select(filter, [$1..300]); # _Robert Israel_, Oct 11 2019
%t A328028 Select[Range[100],!PrimeQ[#]&&!MatchQ[DeleteCases[Divisors[#],1|#],{___,x_,y_,___}/;Divisible[y,x]]&]
%Y A328028 Positions of 0's or 2's in A328026.
%Y A328028 1 and positions of 1's in A328194.
%Y A328028 The version including primes is A328161.
%Y A328028 Partitions with no consecutive divisibilities are A328171.
%Y A328028 Numbers whose proper divisors have no consecutive successions are A088725.
%Y A328028 Contains A001358.
%Y A328028 Cf. A000005, A060680, A060775, A067513, A163870, A328162, A328189.
%K A328028 nonn
%O A328028 1,2
%A A328028 _Gus Wiseman_, Oct 06 2019