|
|
A302590
|
|
Squarefree numbers whose prime indices are prime numbers.
|
|
24
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
COMMENTS
|
A prime index of n is a number m such that prime(m) divides n.
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
|
|
|
FORMULA
|
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.
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|