A325162 Squarefree numbers with no two prime indices differing by less than 3.

1, 2, 3, 5, 7, 11, 13, 14, 17, 19, 22, 23, 26, 29, 31, 33, 34, 37, 38, 39, 41, 43, 46, 47, 51, 53, 57, 58, 59, 61, 62, 65, 67, 69, 71, 73, 74, 79, 82, 83, 85, 86, 87, 89, 93, 94, 95, 97, 101, 103, 106, 107, 109, 111, 113, 115, 118, 119, 122, 123, 127, 129, 131

Offset: 1

Author

Gus Wiseman, Apr 05 2019

Keywords

• nonn

Comments

A prime index of n is a number m such that prime(m) divides n.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions into distinct parts, no two differing by less than 3 (counted by A025157).

Examples

			The sequence of terms together with their prime indices begins:
   1: {}
   2: {1}
   3: {2}
   5: {3}
   7: {4}
  11: {5}
  13: {6}
  14: {1,4}
  17: {7}
  19: {8}
  22: {1,5}
  23: {9}
  26: {1,6}
  29: {10}
  31: {11}
  33: {2,5}
  34: {1,7}
  37: {12}
  38: {1,8}
  39: {2,6}
		

Links

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

Crossrefs

Cf. A001227, A003114, A005117, A025157, A034296, A056239, A073485, A073491, A089995, A112798, A116931, A319630, A325160, A325161.

Programs

Maple

filter:= proc(n) local F;
  F:= ifactors(n)[2];
  if ormap(t -> t[2]>1, F) then return false fi;
  if nops(F) <= 1 then return true fi;
  F:= map(numtheory:-pi,sort(map(t -> t[1],F)));
  min(F[2..-1]-F[1..-2]) >= 3;
end proc:
select(filter, [$1..200]); # Robert Israel, Apr 08 2019

Mathematica

Select[Range[100],Min@@Differences[Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]]>2&]