A302539 Squarefree numbers whose prime indices other than 1 are prime numbers.

1, 2, 3, 5, 6, 10, 11, 15, 17, 22, 30, 31, 33, 34, 41, 51, 55, 59, 62, 66, 67, 82, 83, 85, 93, 102, 109, 110, 118, 123, 127, 134, 155, 157, 165, 166, 170, 177, 179, 186, 187, 191, 201, 205, 211, 218, 241, 246, 249, 254, 255, 277, 283, 295, 310, 314, 327, 330

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

			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}}
15: {{1},{2}}
17: {{4}}
22: {{},{3}}
30: {{},{1},{2}}
31: {{5}}
33: {{1},{3}}
34: {{},{4}}
41: {{6}}
51: {{1},{4}}
55: {{2},{3}}
59: {{7}}
62: {{},{5}}
66: {{},{1},{3}}
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000961, A001222, A003963, A005117, A006450, A007716, A056239, A076610, A275024, A281113, A302242, A302243, A302534.

Programs

Mathematica

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[400],SquareFreeQ[#]&&And@@(PrimeQ/@DeleteCases[primeMS[#],1])&]

PARI

ok(n)={issquarefree(n) && !#select(p->p>2 && !isprime(primepi(p)), factor(n)[,1])} \\ Andrew Howroyd, Aug 26 2018

Formula

Sum_{n>=1} 1/a(n) = (3/2) * Product_{p in A006450} (1 + 1/p) converges since the sum of the reciprocals of A006450 converges. - Amiram Eldar, Feb 02 2021