A329403 Prime numbers p such that the sum of the prime numbers up to its square root equals primepi(p).

11, 29, 59, 179, 389, 541, 5399, 12401, 13441, 40241, 81619, 219647, 439367, 1231547, 1263173, 1279021, 1699627, 1718471, 1756397, 1775903, 2603929, 2675927, 2699911, 2799149, 7580569, 7889627, 8206831, 18398983, 18470987, 34456153, 34660711, 34865977, 40564967, 40677407, 40787531

Offset: 1

Author

Juan Moreno Borrallo, Nov 13 2019

Keywords

• nonn

Comments

There exist infinitely many such prime numbers, as proved by @GH from MO in the link provided to Mathoverflow. - Juan Moreno Borrallo, Mar 15 2021

It follows that the sum of prime numbers up to the square root of n is infinitely often equal to the prime counting function up to n. - Juan Moreno Borrallo, Mar 15 2021

Examples

			The square root of the 5th prime (11) is 3, and the sum of prime numbers up to 3 is 2+3 = 5, so 11 is a term of the sequence.
		

Links

• Juan Moreno Borrallo, An Approximation to the Prime Counting Function Through the Sum of Consecutive Prime Numbers, viXra:1710.0205, 2017.
• Juan Moreno Borrallo, The prime counting function and the sum of prime numbers, viXra:1911.0316, 2019.
• Mathoverflow, Set of prime numbers q such that the sum of the prime numbers p up to its square root equals primepi(q)

Crossrefs

Cf. A000006, A007504, A059396.

Programs

Magma

[NthPrime(k):k in [1..100000]| &+PrimesInInterval(1, Floor(Sqrt(NthPrime(k)))) eq k]; // Marius A. Burtea, Nov 13 2019

Mathematica

Select[Prime@ Range@ PrimePi[10^6], Total@ Prime@ Range@ PrimePi@ Sqrt[#] == PrimePi@ # &] (* Michael De Vlieger, Dec 27 2019 *)

PARI

isok(p) = isprime(p) && (primepi(p) == sum(k=1, sqrtint(p), if (isprime(k), k))); \\ Michel Marcus, Nov 13 2019