A342070 Numbers k such that there are more primes in the interval [2*k+1, 3*k] than there are in the interval [k+1, 2*k].

5, 8, 14, 18, 20, 29, 47, 48, 67, 68, 81, 95, 109, 110, 111, 113, 168, 173, 277, 278, 280, 281, 283, 284, 288, 293, 295, 296, 710, 711, 713, 1323

Offset: 1

Author

Jon E. Schoenfield, Mar 23 2021

Keywords

• nonn
• more

Comments

Conjecture: 1323 is the final term.

If there are at least as many primes in [1, m] as there are in [m+1, 2*m] for all positive integers m, then this sequence consists of the numbers k such that A342068(k)=3.

Examples

			The intervals [1, 100], [101, 200], and [201, 300] contain 25, 21, and 16 primes respectively (cf. A038822); 16 < 21, so 100 is not a term of the sequence.
The intervals [1, 20], [21, 40], and [41, 60] contain 8, 4, and 5 primes, respectively; 5 > 4, so 20 is a term.
		

Crossrefs

Cf. A038822, A342068, A342069, A342071, A342839.

Programs

Python

from sympy import primepi
def ok(n): return primepi(3*n) > 2*primepi(2*n) - primepi(n)
print([m for m in range(9999) if ok(m)]) # Michael S. Branicky, Mar 23 2021