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

1,2

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).

Links

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

Example

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}

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&]

Crossrefs

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

Sequence in context: A371653 A298536 A331784 * A331635 A320055 A072697

Adjacent sequences: A325159 A325160 A325161 * A325163 A325164 A325165

Keyword

nonn

Author

Gus Wiseman, Apr 05 2019

Status

approved