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%I #7 Oct 29 2019 21:08:42
%S 1,2,3,5,7,11,13,15,17,19,23,29,31,33,35,37,41,43,47,51,53,55,59,61,
%T 67,69,71,73,77,79,83,85,89,91,93,95,97,101,103,105,107,109,113,119,
%U 123,127,131,137,139,141,143,145,149,151,155,157,161,163,165,167
%N Numbers whose prime indices have no consecutive divisible parts, meaning no prime index is a divisor of the next-smallest prime index, counted with multiplicity.
%C First differs from A304713 in having 105, with prime indices {2, 3, 4}.
%C A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
%F Intersection of A005117 and A328674.
%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 15: {2,3}
%e 17: {7}
%e 19: {8}
%e 23: {9}
%e 29: {10}
%e 31: {11}
%e 33: {2,5}
%e 35: {3,4}
%e 37: {12}
%e 41: {13}
%e 43: {14}
%e 47: {15}
%e 51: {2,7}
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t Select[Range[100],!MatchQ[primeMS[#],{___,x_,y_,___}/;Divisible[y,x]]&]
%Y A subset of A005117.
%Y These are the Heinz numbers of the partitions counted by A328171.
%Y The non-strict version is A328674 (Heinz numbers for A328675).
%Y The version for relatively prime instead of indivisible is A328335.
%Y Compositions without consecutive divisibilities are A328460.
%Y Numbers whose binary indices lack consecutive divisibilities are A328593.
%Y The version with all pairs indivisible is A304713.
%Y Cf. A056239, A112798, A316476, A318726, A318729, A326704, A328336, A328598, A328599.
%K nonn
%O 1,2
%A _Gus Wiseman_, Oct 26 2019
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