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A302590 Squarefree numbers whose prime indices are prime numbers.

%I A302590 #22 Feb 06 2021 15:56:37
%S A302590 1,3,5,11,15,17,31,33,41,51,55,59,67,83,85,93,109,123,127,155,157,165,
%T A302590 177,179,187,191,201,205,211,241,249,255,277,283,295,327,331,335,341,
%U A302590 353,367,381,401,415,431,451,461,465,471,509,527,537,545,547,561,563
%N A302590 Squarefree numbers whose prime indices are prime numbers.
%C A302590 A prime index of n is a number m such that prime(m) divides n.
%C A302590 From _David A. Corneth_, Feb 05 2021: (Start)
%C A302590 Product_{p in A006450} (p + 1)/p where primepi(p) <= 10^k for k = 3..9 respectively is
%C A302590 2.3221793975627545730894469494385382768...
%C A302590 2.3962097386916566795581118542505513350...
%C A302590 2.4423525010102788492232765893521739629...
%C A302590 2.4739349879225654126399615785205666552...
%C A302590 2.4969363158706022367680967716958174889...
%C A302590 2.5144436325229538304870684054018856517...
%C A302590 2.5282263225826916578696019016723107071... (End)
%H A302590 Andrew Howroyd, <a href="/A302590/b302590.txt">Table of n, a(n) for n = 1..1000</a>
%F A302590 Intersection of A005117 and A076610.
%F A302590 Sum_{n>=1} 1/a(n) = Product_{p in A006450} (1 + 1/p) converges since the sum of the reciprocals of A006450 converges. - _Amiram Eldar_, Feb 02 2021
%e A302590 Entry A302242 describes a correspondence between positive integers and multiset multisystems. In this case it gives the following sequence of set systems.
%e A302590 001: {}
%e A302590 003: {{1}}
%e A302590 005: {{2}}
%e A302590 011: {{3}}
%e A302590 015: {{1},{2}}
%e A302590 017: {{4}}
%e A302590 031: {{5}}
%e A302590 033: {{1},{3}}
%e A302590 041: {{6}}
%e A302590 051: {{1},{4}}
%e A302590 055: {{2},{3}}
%e A302590 059: {{7}}
%e A302590 067: {{8}}
%e A302590 083: {{9}}
%e A302590 085: {{2},{4}}
%e A302590 093: {{1},{5}}
%e A302590 109: {{10}}
%e A302590 123: {{1},{6}}
%e A302590 127: {{11}}
%e A302590 155: {{2},{5}}
%e A302590 157: {{12}}
%e A302590 165: {{1},{2},{3}}
%t A302590 primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A302590 Select[Range[600],SquareFreeQ[#]&&And@@PrimeQ/@primeMS[#]&]
%o A302590 (PARI) ok(n)={issquarefree(n) && !#select(p->!isprime(primepi(p)), factor(n)[,1])} \\ _Andrew Howroyd_, Aug 26 2018
%Y A302590 Cf. A000961, A001222, A003963, A005117, A006450, A007716, A056239, A076610, A275024, A281113, A302242, A302243, A302568.
%K A302590 nonn
%O A302590 1,2
%A A302590 _Gus Wiseman_, Apr 10 2018