A237658 -id:A237658

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A237659

Primes p with pi(p) and pi(p^2) both prime, where pi(.) is given by A000720.

+10
3

17, 41, 59, 109, 127, 157, 353, 367, 709, 1153, 1787, 3319, 3407, 3911, 5851, 6037, 6217, 6469, 8389, 9103, 9319, 10663, 13709, 14107, 14591, 15683, 18433, 19463, 19577, 20107, 21727, 23209, 27809, 29383, 32797, 35023, 36251, 36599, 38351, 39239

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

This is a subsequence of A237658.

Conjecture: The sequence has infinitely many terms.

LINKS

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (n = 1..600 from Zhi-Wei Sun)

EXAMPLE

a(1) = 17 with pi(17) = 7 and pi(17^2) = 61 both prime.

a(2) = 41 with pi(41) = 13 and pi(41^2) = 263 both prime.

MATHEMATICA

p[m_]:=PrimeQ[PrimePi[m^2]]

n=0; Do[If[p[Prime[Prime[k]]], n=n+1; Print[n, " ", Prime[Prime[k]]]], {k, 1, 1000}]

CROSSREFS

Cf. A000040, A000290, A000720, A006450, A038107, A237656, A237657, A237658.

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Feb 11 2014

STATUS

approved

A237687

Primes p with pi(p), pi(pi(p)) and pi(p^2) all prime, where pi(.) is given by A000720.

+10
2

59, 127, 709, 1153, 1787, 9319, 13709, 19577, 32797, 35023, 39239, 40819, 53353, 62921, 75269, 90023, 161159, 191551, 218233, 228451, 235891, 238339, 239087, 272999, 289213, 291619, 339601, 439357, 500741, 513683

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

This is a subsequence of A237659.

Conjecture: The sequence has infinitely many terms.

LINKS

Chai Wah Wu, Table of n, a(n) for n = 1..1000 (n = 1..150 from Zhi-Wei Sun)

EXAMPLE

a(1) = 59 with 59, pi(59) = 17, pi(pi(59)) = pi(17) = 7 and pi(59^2) = 487 all prime.

MATHEMATICA

p[m_]:=PrimeQ[PrimePi[m^2]]

n=0; Do[If[p[Prime[Prime[Prime[k]]]], n=n+1; Print[n, " ", Prime[Prime[Prime[k]]]]], {k, 1, 1000}]

CROSSREFS

Cf. A000040, A000290, A000720, A006450, A038107, A038580, A237656, A237657, A237658, A237659.

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Feb 11 2014

STATUS

approved

A233463

Numbers n such that the three numbers pi(n), pi(n^2), and pi(n^3) are prime.

+10
1

6, 353, 804, 1175, 3482, 3570, 5062, 6217, 10663, 18055, 38712, 42297, 44976, 47626, 48132, 52166, 65611, 67353, 75699, 79864, 85094, 91723, 96057, 99161, 110008, 118551, 125829, 126017, 127286, 132545, 156376, 156694, 159295, 167129, 167366, 170938, 179290

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

pi(k) is the number of primes less than or equal to k.

Next term is greater than 63117 and the Mathematica program given here could not find it.

LINKS

Chai Wah Wu, Table of n, a(n) for n = 1..100

EXAMPLE

6 is in the sequence because the three numbers pi(6)=3, pi(6^2)=11, and pi(6^3)=47 are prime.

MATHEMATICA

Do[If[PrimeQ[PrimePi[m]]&&PrimeQ[PrimePi[m^2]]&&PrimeQ[PrimePi[m^3]], Print[m]], {m, 63117}]

Select[Range[11000], AllTrue[PrimePi[{#, #^2, #^3}], PrimeQ]&] (* The program generates the first 9 terms of the sequence. To generate more, increase the Range constant but the program may take a long time to run. *) (* Harvey P. Dale, Dec 27 2021 *)

PROG

(PARI) isok(n) = isprime(primepi(n)) && isprime(primepi(n^2)) && isprime(primepi(n^3)); \\ Michel Marcus, Apr 28 2018

CROSSREFS

Cf. A000720, A237658.

KEYWORD

nonn

AUTHOR

Farideh Firoozbakht, Feb 11 2014

EXTENSIONS

a(17)-a(37) from Chai Wah Wu, Apr 24 2018

STATUS

approved

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