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

1,2

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)

Links

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

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

Example

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}}

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A153798 A368101 A316092 * A316151 A265121 A281575

Adjacent sequences: A302587 A302588 A302589 * A302591 A302592 A302593

Keyword

nonn

Author

Gus Wiseman, Apr 10 2018

Status

approved