%I #6 Nov 11 2021 15:32:08

%S 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 32,33,34,35,37,38,40,41,43,44,46,47,48,51,52,53,55,56,58,59,60,61,62,

%U 64,66,67,68,69,70,71,73,74,76,77,79,80,82,83,85,86,88,89

%N Numbers that are either prime or whose prime indices are pairwise coprime. Heinz numbers of integer partitions with pairwise coprime parts.

%C A prime index of n is a number m such that prime(m) divides n.

%C The Heinz number of an integer partition (y_1,..,y_k) is prime(y_1)*..*prime(y_k).

%H Charles R Greathouse IV, <a href="/A302569/b302569.txt">Table of n, a(n) for n = 1..10000</a>

%e Entry A302242 describes a correspondence between positive integers and multiset multisystems. In this case it gives the following sequence of multiset systems.

%e 02: {{}}

%e 03: {{1}}

%e 04: {{},{}}

%e 05: {{2}}

%e 06: {{},{1}}

%e 07: {{1,1}}

%e 08: {{},{},{}}

%e 10: {{},{2}}

%e 11: {{3}}

%e 12: {{},{},{1}}

%e 13: {{1,2}}

%e 14: {{},{1,1}}

%e 15: {{1},{2}}

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

%e 17: {{4}}

%e 19: {{1,1,1}}

%e 20: {{},{},{2}}

%e 22: {{},{3}}

%e 23: {{2,2}}

%e 24: {{},{},{},{1}}

%e 26: {{},{1,2}}

%e 28: {{},{},{1,1}}

%e 29: {{1,3}}

%e 30: {{},{1},{2}}

%e 31: {{5}}

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

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

%t Select[Range[200],Or[PrimeQ[#],CoprimeQ@@primeMS[#]]&]

%o (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 Subsequence of A122132.

%Y Cf. A000961, A001222, A005117, A007359, A007716, A051424, A056239, A076610, A101268, A275024, A302505, A302568.

%K nonn

%O 1,1

%A _Gus Wiseman_, Apr 10 2018