A325162 - OEIS

%I #10 Apr 08 2019 11:10:09

%S 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 51,53,57,58,59,61,62,65,67,69,71,73,74,79,82,83,85,86,87,89,93,94,95,

%U 97,101,103,106,107,109,111,113,115,118,119,122,123,127,129,131

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

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

%C 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 Robert Israel, <a href="/A325162/b325162.txt">Table of n, a(n) for n = 1..10000</a>

%e The sequence of terms together with their prime indices begins:

%e    1: {}

%e    2: {1}

%e    3: {2}

%e    5: {3}

%e    7: {4}

%e   11: {5}

%e   13: {6}

%e   14: {1,4}

%e   17: {7}

%e   19: {8}

%e   22: {1,5}

%e   23: {9}

%e   26: {1,6}

%e   29: {10}

%e   31: {11}

%e   33: {2,5}

%e   34: {1,7}

%e   37: {12}

%e   38: {1,8}

%e   39: {2,6}

%p filter:= proc(n) local F;

%p   F:= ifactors(n)[2];

%p   if ormap(t -> t[2]>1, F) then return false fi;

%p   if nops(F) <= 1 then return true fi;

%p   F:= map(numtheory:-pi,sort(map(t -> t[1],F)));

%p   min(F[2..-1]-F[1..-2]) >= 3;

%p end proc:

%p select(filter, [$1..200]); # _Robert Israel_, Apr 08 2019

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

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

%K nonn

%O 1,2

%A _Gus Wiseman_, Apr 05 2019