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A329403
Prime numbers p such that the sum of the prime numbers up to its square root equals primepi(p).
0
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
1,1
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
EXAMPLE
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.
MATHEMATICA
Select[Prime@ Range@ PrimePi[10^6], Total@ Prime@ Range@ PrimePi@ Sqrt[#] == PrimePi@ # &] (* Michael De Vlieger, Dec 27 2019 *)
PROG
(PARI) isok(p) = isprime(p) && (primepi(p) == sum(k=1, sqrtint(p), if (isprime(k), k))); \\ Michel Marcus, Nov 13 2019
(Magma) [NthPrime(k):k in [1..100000]| &+PrimesInInterval(1, Floor(Sqrt(NthPrime(k)))) eq k]; // Marius A. Burtea, Nov 13 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved