A328161 Numbers n that are prime or whose proper divisors (greater than 1 and less than n) have no consecutive divisibilities.

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

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

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.

Cf. A000005, A060680, A060775, A067513, A088725, A163870, A328162.

Programs

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]]&]