A302590 - OEIS

%I #22 Feb 06 2021 15:56:37

%S 1,3,5,11,15,17,31,33,41,51,55,59,67,83,85,93,109,123,127,155,157,165,

%T 177,179,187,191,201,205,211,241,249,255,277,283,295,327,331,335,341,

%U 353,367,381,401,415,431,451,461,465,471,509,527,537,545,547,561,563

%N Squarefree numbers whose prime indices are prime numbers.

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

%C From _David A. Corneth_, Feb 05 2021: (Start)

%C Product_{p in A006450} (p + 1)/p where primepi(p) <= 10^k for k = 3..9 respectively is

%C 2.3221793975627545730894469494385382768...

%C 2.3962097386916566795581118542505513350...

%C 2.4423525010102788492232765893521739629...

%C 2.4739349879225654126399615785205666552...

%C 2.4969363158706022367680967716958174889...

%C 2.5144436325229538304870684054018856517...

%C 2.5282263225826916578696019016723107071... (End)

%H Andrew Howroyd, <a href="/A302590/b302590.txt">Table of n, a(n) for n = 1..1000</a>

%F Intersection of A005117 and A076610.

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

%e 001: {}

%e 003: {{1}}

%e 005: {{2}}

%e 011: {{3}}

%e 015: {{1},{2}}

%e 017: {{4}}

%e 031: {{5}}

%e 033: {{1},{3}}

%e 041: {{6}}

%e 051: {{1},{4}}

%e 055: {{2},{3}}

%e 059: {{7}}

%e 067: {{8}}

%e 083: {{9}}

%e 085: {{2},{4}}

%e 093: {{1},{5}}

%e 109: {{10}}

%e 123: {{1},{6}}

%e 127: {{11}}

%e 155: {{2},{5}}

%e 157: {{12}}

%e 165: {{1},{2},{3}}

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

%t Select[Range[600],SquareFreeQ[#]&&And@@PrimeQ/@primeMS[#]&]

%o (PARI) ok(n)={issquarefree(n) && !#select(p->!isprime(primepi(p)), factor(n)[,1])} \\ _Andrew Howroyd_, Aug 26 2018

%Y Cf. A000961, A001222, A003963, A005117, A006450, A007716, A056239, A076610, A275024, A281113, A302242, A302243, A302568.

%K nonn

%O 1,2

%A _Gus Wiseman_, Apr 10 2018