A302602 Numbers that are powers of a prime number whose prime index is either 1 or also a prime number.

1, 2, 3, 4, 5, 8, 9, 11, 16, 17, 25, 27, 31, 32, 41, 59, 64, 67, 81, 83, 109, 121, 125, 127, 128, 157, 179, 191, 211, 241, 243, 256, 277, 283, 289, 331, 353, 367, 401, 431, 461, 509, 512, 547, 563, 587, 599, 617, 625, 709, 729, 739, 773, 797, 859, 877, 919

Offset: 1

Author

Gus Wiseman, Apr 10 2018

Keywords

• nonn

Comments

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

Examples

			25 is in the sequence because 25 = prime(3)^2 and 3 is a prime number.
Entry A302242 describes a correspondence between positive integers and multiset multisystems. In this case it gives the following sequence of set systems.
01: {}
02: {{}}
03: {{1}}
04: {{},{}}
05: {{2}}
08: {{},{},{}}
09: {{1},{1}}
11: {{3}}
16: {{},{},{},{}}
17: {{4}}
25: {{2},{2}}
27: {{1},{1},{1}}
31: {{5}}
32: {{},{},{},{},{}}
41: {{6}}
59: {{7}}
64: {{},{},{},{},{},{}}
67: {{8}}
81: {{1},{1},{1},{1}}
83: {{9}}
		

Crossrefs

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

Programs

Mathematica

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

PARI

isok(n) = (n==1) || ((isprimepower(n, &p)) && ((p==2) || isprime(primepi(p)))); \\ Michel Marcus, Apr 10 2018