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%I #6 Mar 23 2021 21:32:23
%S 12,19,22,32,42,45,49,50,52,54,57,59,70,71,72,73,74,75,101,102,115,
%T 116,117,121,122,123,124,126,132,143,180,182,184,185,186,187,188,189,
%U 190,192,194,195,197,268,269,309,310,311,312,322,323,325,326,327,328,329
%N Numbers k such that there are more primes in the interval [3*k+1, 4*k] than there are in the interval [2*k+1, 3*k].
%C After a(194)=3977, there are no more terms < 100000.
%C Conjecture: a(194)=3977 is the final term.
%C For each of the first 194 terms k, there are at least as many primes in [1, k] as there are in [k+1, 2*k], and at least as many primes in [k+1, 2*k] as there are in [2*k+1, 3*k], so A342068(k)=4.
%e The intervals [1, 100], [101, 200], [201, 300], and [301, 400] contain 25, 21, 16, and 16 primes respectively (cf. A038822); the 4th interval does not contain more primes than does the 3rd, so 100 is not a term of the sequence.
%e However, the intervals [1, 101], [102, 202], [203, 303], and [304, 404] contain 26, 20, 16, and 17 primes, respectively; 17 > 16, so 101 is a term.
%Y Cf. A342068, A342069, A342070, A342839.
%K nonn
%O 1,1
%A _Jon E. Schoenfield_, Mar 23 2021