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Numbers that are powers of a prime number whose prime index is either 1 or also a prime number.
0

%I #8 Apr 11 2018 02:55:17

%S 1,2,3,4,5,8,9,11,16,17,25,27,31,32,41,59,64,67,81,83,109,121,125,127,

%T 128,157,179,191,211,241,243,256,277,283,289,331,353,367,401,431,461,

%U 509,512,547,563,587,599,617,625,709,729,739,773,797,859,877,919

%N Numbers that are powers of a prime number whose prime index is either 1 or also a prime number.

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

%e 25 is in the sequence because 25 = prime(3)^2 and 3 is a prime number.

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

%e 01: {}

%e 02: {{}}

%e 03: {{1}}

%e 04: {{},{}}

%e 05: {{2}}

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

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

%e 11: {{3}}

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

%e 17: {{4}}

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

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

%e 31: {{5}}

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

%e 41: {{6}}

%e 59: {{7}}

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

%e 67: {{8}}

%e 81: {{1},{1},{1},{1}}

%e 83: {{9}}

%t Select[Range[1000],#===1||MatchQ[FactorInteger[#],{{_?(#===2||PrimeQ[PrimePi[#]]&),_}}]&]

%o (PARI) isok(n) = (n==1) || ((isprimepower(n, &p)) && ((p==2) || isprime(primepi(p)))); \\ _Michel Marcus_, Apr 10 2018

%Y Cf. A000961, A001222, A003963, A005117, A007425, A007716, A056239, A275024, A281113, A295931, A301764, A302242, A302243.

%K nonn

%O 1,2

%A _Gus Wiseman_, Apr 10 2018