A302601 Numbers that are powers of a prime number whose prime index is also a prime power (not including 1).

1, 3, 5, 7, 9, 11, 17, 19, 23, 25, 27, 31, 41, 49, 53, 59, 67, 81, 83, 97, 103, 109, 121, 125, 127, 131, 157, 179, 191, 211, 227, 241, 243, 277, 283, 289, 311, 331, 343, 353, 361, 367, 401, 419, 431, 461, 509, 529, 547, 563, 587, 599, 617, 625, 661, 691, 709

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

			49 is in the sequence because 49 = prime(4)^2 = prime(prime(1)^2)^2.
Entry A302242 describes a correspondence between positive integers and multiset multisystems. In this case it gives the following sequence of multiset multisystems.
001: {}
003: {{1}}
005: {{2}}
007: {{1,1}}
009: {{1},{1}}
011: {{3}}
017: {{4}}
019: {{1,1,1}}
023: {{2,2}}
025: {{2},{2}}
027: {{1},{1},{1}}
031: {{5}}
041: {{6}}
049: {{1,1},{1,1}}
053: {{1,1,1,1}}
059: {{7}}
067: {{8}}
081: {{1},{1},{1},{1}}
083: {{9}}
097: {{3,3}}
103: {{2,2,2}}
109: {{10}}
121: {{3},{3}}
125: {{2},{2},{2}}
127: {{11}}
131: {{1,1,1,1,1}}
		

Crossrefs

Cf. A000961, A001222, A003963, A005117, A007425, A007716, A056239, A275024, A281113, A295931, A301764, A302242, A302243, A302498.

Programs

Mathematica

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

PARI

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