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A342068 a(n) is the smallest k > 1 such that there are more primes in the interval [(k-1)*n + 1, k*n] than there are in the interval [(k-2)*n + 1, (k-1)*n].

%I A342068 #33 Jul 01 2021 06:24:57
%S A342068 2,6,10,5,3,7,5,3,5,5,7,4,11,3,5,5,7,3,4,3,7,4,5,5,5,6,6,9,3,6,8,4,6,
%T A342068 5,7,5,5,6,5,5,7,4,9,6,4,10,3,3,4,4,7,4,6,4,5,5,4,5,4,8,6,7,7,5,10,6,
%U A342068 3,3,6,4,4,4,4,4,4,9,8,6,6,6,3,5,6,5,6,5
%N A342068 a(n) is the smallest k > 1 such that there are more primes in the interval [(k-1)*n + 1, k*n] than there are in the interval [(k-2)*n + 1, (k-1)*n].
%C A342068 a(519) is a noteworthy record high value; a(n) < 13 for all n < 519, and a(n) < 19 for all n < 9363 except that a(519)=19.
%H A342068 Jon E. Schoenfield, <a href="/A342068/b342068.txt">Table of n, a(n) for n = 1..10000</a>
%e A342068 The 1st 100 positive integers,   1..100, include 25 primes;
%e A342068 the 2nd 100 positive integers, 101..200, include 21 primes;
%e A342068 the 3rd 100 positive integers, 201..300, include 16 primes;
%e A342068 the 4th 100 positive integers, 301..400, include 16 primes;
%e A342068 the 5th 100 positive integers, 401..500, include 17 primes.
%e A342068 The sequence 25, 21, 16, 16, 17, is nonincreasing until we reach the 5th term, 17, so a(100) = 5.
%e A342068 Considering the positive integers in consecutive intervals of length 519, instead (i.e., [1,519], [2,1038], [3,1557], ...) and counting the primes in each interval, we get a sequence that is nonincreasing until we reach the 19th term, since the 19th interval, [9343,9861], contains more primes than does the 18th, so a(519)=19.
%p A342068 a:= proc(n) uses numtheory; local i, j, k; i:= n;
%p A342068       for k do j:= pi(k*n)-pi((k-1)*n);
%p A342068         if j>i then break else i:=j fi
%p A342068       od; k
%p A342068     end:
%p A342068 seq(a(n), n=1..100);  # _Alois P. Heinz_, Mar 21 2021
%t A342068 a[n_] := Module[{i = n, j, k},
%t A342068      For[k = 1, True, k++, j = PrimePi[k*n] - PrimePi[(k-1)*n];
%t A342068      If[j > i, Break[], i = j]]; k];
%t A342068 Array[a, 100] (* _Jean-François Alcover_, Jul 01 2021, after _Alois P. Heinz_ *)
%o A342068 (Python)
%o A342068 from sympy import primepi
%o A342068 def A342068(n):
%o A342068     k, a, b, c = 2,0,primepi(n),primepi(2*n)
%o A342068     while a+c <= 2*b:
%o A342068         k += 1
%o A342068         a, b, c = b, c, primepi(k*n)
%o A342068     return k # _Chai Wah Wu_, Mar 25 2021
%Y A342068 Cf. A000040, A342069, A342070, A342071, A342839, A342852.
%K A342068 nonn
%O A342068 1,1
%A A342068 _Jon E. Schoenfield_, Mar 21 2021