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

1, 4, 6, 9, 10, 12, 14, 15, 21, 22, 24, 25, 26, 30, 33, 34, 35, 36, 38, 39, 45, 46, 48, 49, 51, 55, 57, 58, 60, 62, 63, 65, 69, 70, 72, 74, 77, 82, 84, 85, 86, 87, 90, 91, 93, 94, 95, 96, 105, 106, 108, 111, 115, 118, 119, 120, 121, 122, 123, 129, 132, 133, 134

Offset: 1

Author

Gus Wiseman, Oct 06 2019

Keywords

• nonn

Examples

			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.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Positions of 0's or 2's in A328026.

1 and positions of 1's in A328194.

The version including primes is A328161.

Partitions with no consecutive divisibilities are A328171.

Numbers whose proper divisors have no consecutive successions are A088725.

Contains A001358.

Cf. A000005, A060680, A060775, A067513, A163870, A328162, A328189.

Programs

Maple

filter:= proc(n) local D,i;
  if isprime(n) then return false 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..300]); # Robert Israel, Oct 11 2019

Mathematica

Select[Range[100],!PrimeQ[#]&&!MatchQ[DeleteCases[Divisors[#],1|#],{_,x_,y_,_}/;Divisible[y,x]]&]