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%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
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