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Numbers whose distinct prime indices are relatively prime and pairwise indivisible.
9

%I #5 Nov 01 2019 18:41:30

%S 2,4,8,15,16,32,33,35,45,51,55,64,69,75,77,85,93,95,99,119,123,128,

%T 135,141,143,145,153,155,161,165,175,177,187,201,205,207,209,215,217,

%U 219,221,225,245,249,253,255,256,265,275,279,287,291,295,297,309,323

%N Numbers whose distinct prime indices are relatively prime and pairwise indivisible.

%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. Stable numbers are listed in A316476.

%F Intersection of A316476 and A289509.

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

%e 2: {1}

%e 4: {1,1}

%e 8: {1,1,1}

%e 15: {2,3}

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

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

%e 33: {2,5}

%e 35: {3,4}

%e 45: {2,2,3}

%e 51: {2,7}

%e 55: {3,5}

%e 64: {1,1,1,1,1,1}

%e 69: {2,9}

%e 75: {2,3,3}

%e 77: {4,5}

%e 85: {3,7}

%e 93: {2,11}

%e 95: {3,8}

%e 99: {2,2,5}

%e 119: {4,7}

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

%t stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];

%t Select[Range[100],GCD@@primeMS[#]==1&&stableQ[primeMS[#],Divisible]&]

%Y These are the Heinz numbers of the partitions counted by A328676.

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

%Y Partitions whose distinct parts are pairwise indivisible are A305148.

%Y The version for binary indices (instead of prime indices) is A328671.

%Y Cf. A000837, A056239, A112798, A285573, A289508, A303362, A304713, A327393, A328460.

%K nonn

%O 1,1

%A _Gus Wiseman_, Oct 30 2019