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A342068
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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].
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6
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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, 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, 3, 3, 6, 4, 4, 4, 4, 4, 4, 9, 8, 6, 6, 6, 3, 5, 6, 5, 6, 5
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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Jon E. Schoenfield, Table of n, a(n) for n = 1..10000
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EXAMPLE
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The 1st 100 positive integers, 1..100, include 25 primes;
the 2nd 100 positive integers, 101..200, include 21 primes;
the 3rd 100 positive integers, 201..300, include 16 primes;
the 4th 100 positive integers, 301..400, include 16 primes;
the 5th 100 positive integers, 401..500, include 17 primes.
The sequence 25, 21, 16, 16, 17, is nonincreasing until we reach the 5th term, 17, so a(100) = 5.
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.
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MAPLE
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a:= proc(n) uses numtheory; local i, j, k; i:= n;
for k do j:= pi(k*n)-pi((k-1)*n);
if j>i then break else i:=j fi
od; k
end:
seq(a(n), n=1..100); # Alois P. Heinz, Mar 21 2021
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MATHEMATICA
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a[n_] := Module[{i = n, j, k},
For[k = 1, True, k++, j = PrimePi[k*n] - PrimePi[(k-1)*n];
If[j > i, Break[], i = j]]; k];
Array[a, 100] (* Jean-François Alcover, Jul 01 2021, after Alois P. Heinz *)
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PROG
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(Python)
from sympy import primepi
def A342068(n):
k, a, b, c = 2, 0, primepi(n), primepi(2*n)
while a+c <= 2*b:
k += 1
a, b, c = b, c, primepi(k*n)
return k # Chai Wah Wu, Mar 25 2021
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CROSSREFS
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Cf. A000040, A342069, A342070, A342071, A342839, A342852.
Sequence in context: A320383 A073662 A086553 * A004055 A077933 A144762
Adjacent sequences: A342065 A342066 A342067 * A342069 A342070 A342071
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KEYWORD
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nonn
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AUTHOR
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Jon E. Schoenfield, Mar 21 2021
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STATUS
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approved
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