A302497 Powers of primes of squarefree index.

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, 83, 101, 109, 113, 121, 125, 127, 128, 137, 139, 149, 157, 163, 167, 169, 179, 181, 191, 199, 211, 233, 241, 243, 256, 257, 269, 271, 277, 283, 289, 293, 313, 317, 331

Offset: 1

Author

Gus Wiseman, Apr 09 2018

Keywords

• nonn

Comments

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

Examples

			49 is not in the sequence because 49 = prime(4)^2 but 4 is not squarefree.
Entry A302242 describes a correspondence between positive integers and multiset multisystems. In this case it gives the following sequence of constant set multisystems.
01: {}
02: {{}}
03: {{1}}
04: {{},{}}
05: {{2}}
08: {{},{},{}}
09: {{1},{1}}
11: {{3}}
13: {{1,2}}
16: {{},{},{},{}}
17: {{4}}
25: {{2},{2}}
27: {{1},{1},{1}}
29: {{1,3}}
31: {{5}}
32: {{},{},{},{},{}}
41: {{6}}
43: {{1,4}}
47: {{2,3}}
59: {{7}}
64: {{},{},{},{},{},{}}
		

Crossrefs

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

Programs

Mathematica

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

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