cp's OEIS Frontend

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

%I A329403 #52 Sep 08 2022 08:46:24
%S A329403 11,29,59,179,389,541,5399,12401,13441,40241,81619,219647,439367,
%T A329403 1231547,1263173,1279021,1699627,1718471,1756397,1775903,2603929,
%U A329403 2675927,2699911,2799149,7580569,7889627,8206831,18398983,18470987,34456153,34660711,34865977,40564967,40677407,40787531
%N A329403 Prime numbers p such that the sum of the prime numbers up to its square root equals primepi(p).
%C A329403 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
%C A329403 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
%H A329403 Juan Moreno Borrallo, <a href="http://vixra.org/abs/1710.0205">An Approximation to the Prime Counting Function Through the Sum of Consecutive Prime Numbers</a>, viXra:1710.0205, 2017.
%H A329403 Juan Moreno Borrallo, <a href="http://vixra.org/abs/1911.0316">The prime counting function and the sum of prime numbers</a>, viXra:1911.0316, 2019.
%H A329403 Mathoverflow, <a href="https://mathoverflow.net/questions/383610/set-of-prime-numbers-q-such-that-sum-limits-p-leq-sqrtqp-piq-where">Set of prime numbers q such that the sum of the prime numbers p up to its square root equals primepi(q)</a>
%e A329403 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.
%t A329403 Select[Prime@ Range@ PrimePi[10^6], Total@ Prime@ Range@ PrimePi@ Sqrt[#] == PrimePi@ # &] (* _Michael De Vlieger_, Dec 27 2019 *)
%o A329403 (PARI) isok(p) = isprime(p) && (primepi(p) == sum(k=1, sqrtint(p), if (isprime(k), k))); \\ _Michel Marcus_, Nov 13 2019
%o A329403 (Magma) [NthPrime(k):k in [1..100000]| &+PrimesInInterval(1, Floor(Sqrt(NthPrime(k)))) eq k]; // _Marius A. Burtea_, Nov 13 2019
%Y A329403 Cf. A000006, A007504, A059396.
%K A329403 nonn
%O A329403 1,1
%A A329403 _Juan Moreno Borrallo_, Nov 13 2019