A342068 - OEIS

%I #33 Jul 01 2021 06:24:57

%S 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 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 3,3,6,4,4,4,4,4,4,9,8,6,6,6,3,5,6,5,6,5

%N 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 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 Jon E. Schoenfield, <a href="/A342068/b342068.txt">Table of n, a(n) for n = 1..10000</a>

%e The 1st 100 positive integers,   1..100, include 25 primes;

%e the 2nd 100 positive integers, 101..200, include 21 primes;

%e the 3rd 100 positive integers, 201..300, include 16 primes;

%e the 4th 100 positive integers, 301..400, include 16 primes;

%e the 5th 100 positive integers, 401..500, include 17 primes.

%e The sequence 25, 21, 16, 16, 17, is nonincreasing until we reach the 5th term, 17, so a(100) = 5.

%e 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 a:= proc(n) uses numtheory; local i, j, k; i:= n;

%p       for k do j:= pi(k*n)-pi((k-1)*n);

%p         if j>i then break else i:=j fi

%p       od; k

%p     end:

%p seq(a(n), n=1..100);  # _Alois P. Heinz_, Mar 21 2021

%t a[n_] := Module[{i = n, j, k},

%t      For[k = 1, True, k++, j = PrimePi[k*n] - PrimePi[(k-1)*n];

%t      If[j > i, Break[], i = j]]; k];

%t Array[a, 100] (* _Jean-François Alcover_, Jul 01 2021, after _Alois P. Heinz_ *)

%o (Python)

%o from sympy import primepi

%o def A342068(n):

%o     k, a, b, c = 2,0,primepi(n),primepi(2*n)

%o     while a+c <= 2*b:

%o         k += 1

%o         a, b, c = b, c, primepi(k*n)

%o     return k # _Chai Wah Wu_, Mar 25 2021

%Y Cf. A000040, A342069, A342070, A342071, A342839, A342852.

%K nonn

%O 1,1

%A _Jon E. Schoenfield_, Mar 21 2021