%I #7 Nov 01 2019 18:40:40

%S 1,2,3,4,5,7,8,9,11,13,15,16,17,19,23,25,27,29,31,32,33,35,37,41,43,

%T 45,47,49,51,53,55,59,61,64,67,69,71,73,75,77,79,81,83,85,89,91,93,95,

%U 97,99,101,103,105,107,109,113,119,121,123,125,127,128,131,135

%N Numbers whose distinct prime indices have no consecutive divisible parts.

%C First differs from A316476 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.

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

%e 1: {}

%e 2: {1}

%e 3: {2}

%e 4: {1,1}

%e 5: {3}

%e 7: {4}

%e 8: {1,1,1}

%e 9: {2,2}

%e 11: {5}

%e 13: {6}

%e 15: {2,3}

%e 16: {1,1,1,1}

%e 17: {7}

%e 19: {8}

%e 23: {9}

%e 25: {3,3}

%e 27: {2,2,2}

%e 29: {10}

%e 31: {11}

%e 32: {1,1,1,1,1}

%e For example, 45 is in the sequence because its distinct prime indices are {2,3} and 2 is not a divisor of 3.

%t Select[Range[100],!MatchQ[PrimePi/@First/@FactorInteger[#],{___,x_,y_,___}/;Divisible[y,x]]&]

%Y These are the Heinz numbers of the partitions counted by A328675.

%Y The strict version is A328603.

%Y Partitions without consecutive divisibilities are A328171.

%Y Compositions without consecutive divisibilities are A328460.

%Y Cf. A056239, A112798, A316476, A318729, A328335, A328336, A328593, A328598.

%K nonn

%O 1,2

%A _Gus Wiseman_, Oct 29 2019