A302590 Squarefree numbers whose prime indices are prime numbers.

1, 3, 5, 11, 15, 17, 31, 33, 41, 51, 55, 59, 67, 83, 85, 93, 109, 123, 127, 155, 157, 165, 177, 179, 187, 191, 201, 205, 211, 241, 249, 255, 277, 283, 295, 327, 331, 335, 341, 353, 367, 381, 401, 415, 431, 451, 461, 465, 471, 509, 527, 537, 545, 547, 561, 563

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.

From David A. Corneth, Feb 05 2021: (Start)

Product_{p in A006450} (p + 1)/p where primepi(p) <= 10^k for k = 3..9 respectively is

2.3221793975627545730894469494385382768...

2.3962097386916566795581118542505513350...

2.4423525010102788492232765893521739629...

2.4739349879225654126399615785205666552...

2.4969363158706022367680967716958174889...

2.5144436325229538304870684054018856517...

2.5282263225826916578696019016723107071... (End)

Examples

			Entry A302242 describes a correspondence between positive integers and multiset multisystems. In this case it gives the following sequence of set systems.
001: {}
003: {{1}}
005: {{2}}
011: {{3}}
015: {{1},{2}}
017: {{4}}
031: {{5}}
033: {{1},{3}}
041: {{6}}
051: {{1},{4}}
055: {{2},{3}}
059: {{7}}
067: {{8}}
083: {{9}}
085: {{2},{4}}
093: {{1},{5}}
109: {{10}}
123: {{1},{6}}
127: {{11}}
155: {{2},{5}}
157: {{12}}
165: {{1},{2},{3}}
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000961, A001222, A003963, A005117, A006450, A007716, A056239, A076610, A275024, A281113, A302242, A302243, A302568.

Programs

Mathematica

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[600],SquareFreeQ[#]&&And@@PrimeQ/@primeMS[#]&]

PARI

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

Formula

Intersection of A005117 and A076610.

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