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

%I A325162 #10 Apr 08 2019 11:10:09
%S A325162 1,2,3,5,7,11,13,14,17,19,22,23,26,29,31,33,34,37,38,39,41,43,46,47,
%T A325162 51,53,57,58,59,61,62,65,67,69,71,73,74,79,82,83,85,86,87,89,93,94,95,
%U A325162 97,101,103,106,107,109,111,113,115,118,119,122,123,127,129,131
%N A325162 Squarefree numbers with no two prime indices differing by less than 3.
%C A325162 A prime index of n is a number m such that prime(m) divides n.
%C A325162 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).
%H A325162 Robert Israel, <a href="/A325162/b325162.txt">Table of n, a(n) for n = 1..10000</a>
%e A325162 The sequence of terms together with their prime indices begins:
%e A325162    1: {}
%e A325162    2: {1}
%e A325162    3: {2}
%e A325162    5: {3}
%e A325162    7: {4}
%e A325162   11: {5}
%e A325162   13: {6}
%e A325162   14: {1,4}
%e A325162   17: {7}
%e A325162   19: {8}
%e A325162   22: {1,5}
%e A325162   23: {9}
%e A325162   26: {1,6}
%e A325162   29: {10}
%e A325162   31: {11}
%e A325162   33: {2,5}
%e A325162   34: {1,7}
%e A325162   37: {12}
%e A325162   38: {1,8}
%e A325162   39: {2,6}
%p A325162 filter:= proc(n) local F;
%p A325162   F:= ifactors(n)[2];
%p A325162   if ormap(t -> t[2]>1, F) then return false fi;
%p A325162   if nops(F) <= 1 then return true fi;
%p A325162   F:= map(numtheory:-pi,sort(map(t -> t[1],F)));
%p A325162   min(F[2..-1]-F[1..-2]) >= 3;
%p A325162 end proc:
%p A325162 select(filter, [$1..200]); # _Robert Israel_, Apr 08 2019
%t A325162 Select[Range[100],Min@@Differences[Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]]>2&]
%Y A325162 Cf. A001227, A003114, A005117, A025157, A034296, A056239, A073485, A073491, A089995, A112798, A116931, A319630, A325160, A325161.
%K A325162 nonn
%O A325162 1,2
%A A325162 _Gus Wiseman_, Apr 05 2019