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Powers of primes of squarefree index.
1

%I #6 Apr 10 2018 21:48:17

%S 1,2,3,4,5,8,9,11,13,16,17,25,27,29,31,32,41,43,47,59,64,67,73,79,81,

%T 83,101,109,113,121,125,127,128,137,139,149,157,163,167,169,179,181,

%U 191,199,211,233,241,243,256,257,269,271,277,283,289,293,313,317,331

%N Powers of primes of squarefree index.

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

%e 49 is not in the sequence because 49 = prime(4)^2 but 4 is not squarefree.

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

%e 01: {}

%e 02: {{}}

%e 03: {{1}}

%e 04: {{},{}}

%e 05: {{2}}

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

%e 09: {{1},{1}}

%e 11: {{3}}

%e 13: {{1,2}}

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

%e 17: {{4}}

%e 25: {{2},{2}}

%e 27: {{1},{1},{1}}

%e 29: {{1,3}}

%e 31: {{5}}

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

%e 41: {{6}}

%e 43: {{1,4}}

%e 47: {{2,3}}

%e 59: {{7}}

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

%t Select[Range[100],Or[#===1,PrimePowerQ[#]&&And@@SquareFreeQ/@PrimePi/@FactorInteger[#][[All,1]]]&]

%o (PARI) is(n) = if(n==1, return(1), my(x=isprimepower(n)); if(x > 0, if(issquarefree(primepi(ceil(n^(1/x)))), return(1)))); 0 \\ _Felix Fröhlich_, Apr 10 2018

%Y Cf. A000961, A001222, A003963, A005117, A007716, A056239, A275024, A279788, A281113, A296132, A301768, A302242, A302243, A302478, A302494.

%K nonn

%O 1,2

%A _Gus Wiseman_, Apr 09 2018