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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.

%I A328161 #15 Oct 30 2019 00:56:33
%S A328161 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,
%T A328161 33,34,35,36,37,38,39,41,43,45,46,47,48,49,51,53,55,57,58,59,60,61,62,
%U A328161 63,65,67,69,70,71,72,73,74,77,79,82,83,84,85,86,87,89,90,91
%N A328161 Numbers n that are prime or whose proper divisors (greater than 1 and less than n) have no consecutive divisibilities.
%H A328161 Robert Israel, <a href="/A328161/b328161.txt">Table of n, a(n) for n = 1..10000</a>
%e A328161 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 A328161 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 A328161 filter:= proc(n) local D,i;
%p A328161   if isprime(n) then return true fi;
%p A328161   D:= sort(convert(numtheory:-divisors(n) minus {1,n}, list));
%p A328161   for i from 1 to nops(D)-1 do if (D[i+1]/D[i])::integer then return false fi od:
%p A328161   true
%p A328161 end proc:
%p A328161 select(filter, [$1..100]); # _Robert Israel_, Oct 11 2019
%t A328161 Select[Range[100],!MatchQ[DeleteCases[Divisors[#],1|#],{___,x_,y_,___}/;Divisible[y,x]]&]
%Y A328161 Equals the union of A328028 and A000040.
%Y A328161 Complement of A328189.
%Y A328161 One, primes, and positions of 1's in A328194.
%Y A328161 Partitions with no consecutive divisibilities are A328171.
%Y A328161 Cf. A000005, A060680, A060775, A067513, A088725, A163870, A328162.
%K A328161 nonn
%O A328161 1,2
%A A328161 _Gus Wiseman_, Oct 06 2019