A237658 -id:A237658 - cp's OEIS frontend

A237659 Primes p with pi(p) and pi(p^2) both prime, where pi(.) is given by A000720.

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

Offset: 1

Zhi-Wei Sun, Feb 11 2014

nonn

This is a subsequence of A237658.

Conjecture: The sequence has infinitely many terms.

			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.

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

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

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}]
Select[Prime[Range[4500]],AllTrue[{PrimePi[#],PrimePi[#^2]},PrimeQ]&] (* Harvey P. Dale, May 10 2025 *)

A237687 Primes p with pi(p), pi(pi(p)) and pi(p^2) all prime, where pi(.) is given by A000720.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Feb 11 2014

nonn

This is a subsequence of A237659.

Conjecture: The sequence has infinitely many terms.

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

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

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

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}]

A233463 Numbers n such that the three numbers pi(n), pi(n^2), and pi(n^3) are prime.

Original entry on oeis.org

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

Offset: 1

Farideh Firoozbakht, Feb 11 2014

nonn

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.

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

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

Cf. A000720, A237658.

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 *)

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

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

cp's OEIS Frontend

A237659 Primes p with pi(p) and pi(p^2) both prime, where pi(.) is given by A000720.

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A237687 Primes p with pi(p), pi(pi(p)) and pi(p^2) all prime, where pi(.) is given by A000720.

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Author

Keywords

Comments

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Mathematica

A233463 Numbers n such that the three numbers pi(n), pi(n^2), and pi(n^3) are prime.

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