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Numbers whose consecutive prime indices are relatively prime.
15

%I #7 Oct 16 2019 08:45:20

%S 1,2,3,4,5,6,7,8,10,11,12,13,14,15,16,17,19,20,22,23,24,26,28,29,30,

%T 31,32,33,34,35,37,38,40,41,43,44,46,47,48,51,52,53,55,56,58,59,60,61,

%U 62,64,66,67,68,69,70,71,73,74,76,77,79,80,82,83,85,86,88

%N Numbers whose consecutive prime indices are relatively prime.

%C First differs from A302569 in having 105, which has 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.

%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 partitions whose consecutive parts are relatively prime (A328172).

%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 6: {1,2}

%e 7: {4}

%e 8: {1,1,1}

%e 10: {1,3}

%e 11: {5}

%e 12: {1,1,2}

%e 13: {6}

%e 14: {1,4}

%e 15: {2,3}

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

%e 17: {7}

%e 19: {8}

%e 20: {1,1,3}

%e 22: {1,5}

%e 23: {9}

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t Select[Range[100],!MatchQ[primeMS[#],{___,x_,y_,___}/;GCD[x,y]>1]&]

%Y A superset of A302569.

%Y Numbers whose prime indices are relatively prime are A289509.

%Y Numbers with no consecutive prime indices relatively prime are A328336.

%Y Cf. A000837, A056239, A112798, A281116, A289508, A318981, A328168, A328169, A328172, A328187, A328188, A328220.

%K nonn

%O 1,2

%A _Gus Wiseman_, Oct 14 2019