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%I A302569 #6 Nov 11 2021 15:32:08
%S A302569 2,3,4,5,6,7,8,10,11,12,13,14,15,16,17,19,20,22,23,24,26,28,29,30,31,
%T A302569 32,33,34,35,37,38,40,41,43,44,46,47,48,51,52,53,55,56,58,59,60,61,62,
%U A302569 64,66,67,68,69,70,71,73,74,76,77,79,80,82,83,85,86,88,89
%N A302569 Numbers that are either prime or whose prime indices are pairwise coprime. Heinz numbers of integer partitions with pairwise coprime parts.
%C A302569 A prime index of n is a number m such that prime(m) divides n.
%C A302569 The Heinz number of an integer partition (y_1,..,y_k) is prime(y_1)*..*prime(y_k).
%H A302569 Charles R Greathouse IV, <a href="/A302569/b302569.txt">Table of n, a(n) for n = 1..10000</a>
%e A302569 Entry A302242 describes a correspondence between positive integers and multiset multisystems. In this case it gives the following sequence of multiset systems.
%e A302569 02: {{}}
%e A302569 03: {{1}}
%e A302569 04: {{},{}}
%e A302569 05: {{2}}
%e A302569 06: {{},{1}}
%e A302569 07: {{1,1}}
%e A302569 08: {{},{},{}}
%e A302569 10: {{},{2}}
%e A302569 11: {{3}}
%e A302569 12: {{},{},{1}}
%e A302569 13: {{1,2}}
%e A302569 14: {{},{1,1}}
%e A302569 15: {{1},{2}}
%e A302569 16: {{},{},{},{}}
%e A302569 17: {{4}}
%e A302569 19: {{1,1,1}}
%e A302569 20: {{},{},{2}}
%e A302569 22: {{},{3}}
%e A302569 23: {{2,2}}
%e A302569 24: {{},{},{},{1}}
%e A302569 26: {{},{1,2}}
%e A302569 28: {{},{},{1,1}}
%e A302569 29: {{1,3}}
%e A302569 30: {{},{1},{2}}
%e A302569 31: {{5}}
%e A302569 32: {{},{},{},{},{}}
%t A302569 primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A302569 Select[Range[200],Or[PrimeQ[#],CoprimeQ@@primeMS[#]]&]
%o A302569 (PARI) is(n)=if(n<9, return(n>1)); n>>=valuation(n,2); if(n<9, return(1)); my(f=factor(n)); if(vecmax(f[,2])>1, return(0)); if(#f~==1, return(1)); my(v=apply(primepi, f[,1]),P=vecprod(v)); for(i=1,#v, if(gcd(v[i],P/v[i])>1, return(0))); 1 \\ _Charles R Greathouse IV_, Nov 11 2021
%Y A302569 Subsequence of A122132.
%Y A302569 Cf. A000961, A001222, A005117, A007359, A007716, A051424, A056239, A076610, A101268, A275024, A302505, A302568.
%K A302569 nonn
%O A302569 1,1
%A A302569 _Gus Wiseman_, Apr 10 2018

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