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A325160 Products of distinct, non-consecutive primes. Squarefree numbers not divisible by any two consecutive primes.

%I A325160 #20 Sep 24 2022 05:47:25
%S A325160 1,2,3,5,7,10,11,13,14,17,19,21,22,23,26,29,31,33,34,37,38,39,41,43,
%T A325160 46,47,51,53,55,57,58,59,61,62,65,67,69,71,73,74,79,82,83,85,86,87,89,
%U A325160 91,93,94,95,97,101,103,106,107,109,110,111,113,115,118,119
%N A325160 Products of distinct, non-consecutive primes. Squarefree numbers not divisible by any two consecutive primes.
%C A325160 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 non-consecutive parts (counted by A003114). The nonsquarefree case is A319630, which gives the Heinz numbers of integer partitions with no consecutive parts (counted by A116931).
%C A325160 The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 6, 52, 515, 5146, 51435, 514416, 5144232, 51442384, ... . Apparently, the asymptotic density of this sequence exists and equals 0.51442... . - _Amiram Eldar_, Sep 24 2022
%H A325160 Amiram Eldar, <a href="/A325160/b325160.txt">Table of n, a(n) for n = 1..10000</a>
%e A325160 The sequence of terms together with their prime indices begins:
%e A325160    1: {}
%e A325160    2: {1}
%e A325160    3: {2}
%e A325160    5: {3}
%e A325160    7: {4}
%e A325160   10: {1,3}
%e A325160   11: {5}
%e A325160   13: {6}
%e A325160   14: {1,4}
%e A325160   17: {7}
%e A325160   19: {8}
%e A325160   21: {2,4}
%e A325160   22: {1,5}
%e A325160   23: {9}
%e A325160   26: {1,6}
%e A325160   29: {10}
%e A325160   31: {11}
%e A325160   33: {2,5}
%e A325160   34: {1,7}
%e A325160   37: {12}
%t A325160 Select[Range[100],Min@@Differences[Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]]>1&]
%o A325160 (PARI) isok(k) = {if (issquarefree(k), my(v = apply(primepi, factor(k)[,1])); ! #select(x->(v[x+1]-v[x] == 1), [1..#v-1]));} \\ _Michel Marcus_, Jan 09 2021
%Y A325160 Cf. A001227, A003114, A005117, A025157, A034296, A056239, A073485, A073491, A089995, A112798, A116931, A319630, A325161, A325162.
%K A325160 nonn
%O A325160 1,2
%A A325160 _Gus Wiseman_, Apr 05 2019