A301766 -id:A301766 - cp's OEIS frontend

A302597 Squarefree numbers whose prime indices are powers of a common prime number.

1, 2, 3, 5, 6, 7, 10, 11, 14, 17, 19, 21, 22, 23, 31, 34, 38, 41, 42, 46, 53, 57, 59, 62, 67, 82, 83, 97, 103, 106, 109, 114, 115, 118, 127, 131, 133, 134, 157, 159, 166, 179, 191, 194, 206, 211, 218, 227, 230, 241, 254, 262, 266, 277, 283, 311, 314, 318, 331

Offset: 1

Gus Wiseman, Apr 10 2018

nonn

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

			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}}
05: {{2}}
06: {{},{1}}
07: {{1,1}}
10: {{},{2}}
11: {{3}}
14: {{},{1,1}}
17: {{4}}
19: {{1,1,1}}
21: {{1},{1,1}}
22: {{},{3}}
23: {{2,2}}
31: {{5}}
34: {{},{4}}
38: {{},{1,1,1}}

Cf. A000961, A001222, A003963, A005117, A007716, A056239, A275024, A047966, A281113, A296134, A301766, A302242, A302243.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],SameQ@@Join@@primeMS/@primeMS[#]&&SquareFreeQ[#]&]

A302600 1, 2, prime numbers of prime index, and prime numbers of prime index times 2.

Original entry on oeis.org

1, 2, 3, 5, 6, 10, 11, 17, 22, 31, 34, 41, 59, 62, 67, 82, 83, 109, 118, 127, 134, 157, 166, 179, 191, 211, 218, 241, 254, 277, 283, 314, 331, 353, 358, 367, 382, 401, 422, 431, 461, 482, 509, 547, 554, 563, 566, 587, 599, 617, 662, 706, 709, 734, 739, 773

Offset: 1

Gus Wiseman, Apr 10 2018

nonn

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

Also squarefree numbers whose prime indices other than 1 are equal prime numbers.

			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}}
05: {{2}}
06: {{},{1}}
10: {{},{2}}
11: {{3}}
17: {{4}}
22: {{},{3}}
31: {{5}}
34: {{},{4}}
41: {{6}}
59: {{7}}
62: {{},{5}}
67: {{8}}
82: {{},{6}}
83: {{9}}

Cf. A000961, A001222, A003963, A005117, A007716, A056239, A275024, A047966, A281113, A296134, A301766, A302242, A302243.

Select[Range[1000],SquareFreeQ[#]&&SameQ@@DeleteCases[primeMS[#],1]&&And@@PrimeQ/@DeleteCases[primeMS[#],1]&]

cp's OEIS Frontend

A302597 Squarefree numbers whose prime indices are powers of a common prime number.

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