A294509 a(n) is the least value of pi(n*m) - pi(n)*pi(m) for any positive m <= n.

0, 1, 0, 1, 0, 1, -1, 0, 1, 2, 1, 1, 0, 1, 2, 3, 1, 2, 0, 1, 2, 2, 1, 2, 3, 3, 4, 5, 3, 4, 2, 2, 3, 4, 5, 6, 5, 6, 6, 6, 4, 4, 3, 4, 4, 5, 4, 4, 4, 5, 6, 6, 5, 5, 6, 7, 7, 8, 6, 7, 6, 6, 6, 7, 8, 9, 8, 8, 8, 8, 7, 7, 5, 5, 6, 7, 8, 9, 7, 8, 9, 9, 7, 8, 8, 9, 9, 10, 8, 9, 10, 10, 11, 12, 13, 13, 11, 12, 12, 12, 10, 10, 9, 10, 11

Offset: 1

Author

Jonathan Sondow and Robert G. Wilson v, Nov 06 2017

Keywords

• sign

Comments

Least value in the n-th row of the table in A294508.

First occurrence of -1, 0, 1, 2, etc. occurs at n = 7, 1, 2, 10, 16, 27, 28, 36, 56, 58, 66, 88, 93, 94, 95, 125, 130, 145, 147, 148, 156, 190, 206, 207, 215, 216, 218, etc.

Last occurrence of -1, 0, 1, 2, etc. occurs at n = 7, 19, 23, 32, 43, 49, 74, 75, 83, 115, 116, 117, 119, 139, 140, 143, 152, 199, 200, 202, 204, 205, 213, 242, 244, 284, 285, etc.

Conjecture: a(n) <= pi(n*m) - pi(n)*pi(m) for all m > n if n <> 5.

Examples

			a(13) = 0 since 0 is the least value in the 13th row of A294508.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..5000

Crossrefs

Cf. A000720, A291440, A294508.

Programs

Maple

f:= n -> min(seq(numtheory:-pi(n*m)-numtheory:-pi(n)*numtheory:-pi(m), m=1..n)):
map(f, [$1..200]); # Robert Israel, Nov 08 2017

Mathematica

t[n_, m_] := PrimePi[n*m] - PrimePi[n]*PrimePi[m]; Min @@@ Table[ t[n, m], {n, 100}, {m, n}]

PARI

a(n) = vecmin(vector(n, m, primepi(n*m) - primepi(n)*primepi(m))); \\ Michel Marcus, Nov 08 2017

Formula

a(n) = min_{1<=m<=n} A294508(n*(n-1)/2 + m).

a(n) <= A291440(n).