A237595 -id:A237595 - cp's OEIS frontend

A237658 Positive integers m with pi(m) and pi(m^2) both prime, where pi(.) is given by A000720.

6, 17, 33, 34, 41, 59, 60, 69, 109, 110, 111, 127, 157, 161, 246, 287, 335, 353, 367, 368, 404, 600, 709, 711, 713, 718, 740, 779, 804, 1153, 1162, 1175, 1437, 1472, 1500, 1526, 1527, 1679, 1729, 1742, 1787, 1826, 2028, 2082, 2104, 2223, 2422, 2616, 2649, 2651

Offset: 1

Zhi-Wei Sun, Feb 10 2014

nonn

The conjecture in A237657 implies that this sequence has infinitely many terms.

For primes in this sequence, see A237659.

			a(1) = 6 since pi(6) = 3 and pi(6^2) = 11 are both prime, but none of pi(1) = 0, pi(2) = 1, pi(3^2) = 4, pi(4^2) = 6 and pi(5^2) = 9 is prime.

Chai Wah Wu, Table of n, a(n) for n = 1..10001 (n = 1..3000 from Zhi-Wei Sun)

Cf. A000040, A000290, A038107, A237595, A237656, A237657, A237659.

p[m_]:=PrimeQ[PrimePi[m]]&&PrimeQ[PrimePi[m^2]]
n=0;Do[If[p[m],n=n+1;Print[n," ",m]],{m,1,1000}]

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

A238570 a(n) = |{0 < k < n: pi((k+1)^2) - pi(k^2) and pi(n^2) - pi(k^2) are both prime}|, where pi(x) denotes the number of primes not exceeding x.

Original entry on oeis.org

0, 1, 1, 1, 3, 4, 2, 1, 2, 3, 1, 1, 3, 3, 1, 1, 3, 2, 2, 2, 4, 5, 2, 5, 3, 6, 4, 5, 4, 1, 2, 2, 6, 4, 2, 1, 3, 1, 1, 5, 5, 1, 6, 3, 3, 7, 4, 6, 1, 4, 5, 3, 4, 4, 7, 6, 4, 7, 6, 6, 1, 3, 3, 5, 6, 6, 3, 4, 9, 6, 4, 2, 5, 3, 8, 3, 3, 6, 8, 6

Offset: 1

Zhi-Wei Sun, Feb 28 2014

nonn

Conjecture: (i) a(n) > 0 for all n > 1.
(ii) If n > 4, then pi(n^2) + pi(k^2) is prime for some k = 2, ..., n-1.
(iii) If n > 0 is not a divisor of 12, then n^2 + pi(k^2) is prime for some k = 2, ..., n-1.

			a(8) = 1 since pi(8^2) - pi(7^2) = 18 - 15 = 3 is prime.
a(61) = 1 since pi(27^2) - pi(26^2) = 129 - 122 = 7 and pi(61^2) - pi(26^2) = 519 - 122 = 397 are both prime.
a(86) = 1 since pi(3^2) - pi(2^2) = 4 - 2 = 2 and pi(86^2) - pi(2^2) = 939 - 2 = 937 are both prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..5000
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014.

Cf. A000040, A000720, A237657, A237595, A238568.

p[k_,n_]:=PrimeQ[PrimePi[(k+1)^2]-PrimePi[k^2]]&&PrimeQ[PrimePi[n^2]-PrimePi[k^2]]
a[n_]:=Sum[If[p[k,n],1,0],{k,1,n-1}]
Table[a[n],{n,1,80}]

cp's OEIS Frontend

A237658 Positive integers m with pi(m) and pi(m^2) both prime, where pi(.) is given by A000720.

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