OFFSET
1,4
COMMENTS
Inspired by A291440.
Mincu and Panaitopol (2008) prove that pi(m*n) >= pi(m)*i(n) for all positive m and n except for m = 5, n = 7; m = 7, n = 5; and m = n = 7.
a(i) = -1 for i = 26 and 28, when n = 7 and m = either 5 or 7.
a(i) = 0 for i = 1, 6, 13, 15, 24, 33, 35, 81, 83, 85, 174, 176, 178; when n=m=1; n=m=3; n=5 and m is either 3 or 5; n=7 and m=3; n=8 and m is either 5 or 7; n=13 and m is either 3, 5, or 7; and n=19 with m being either 3, 5 or 7.
First occurrence of k = -1, 0, 1, 2, .., 20, 21, etc. occurs at i = 26, 1, 2, 4, 11, 22, 51, 45, 77, 54, 55, 76, 115, 120, 130, 187, 168, 135, 171, 136, 169, 274, etc.
Last occurrence of k >= -1 occurs at i = 28, 178, 260, 499, 906, 1179, 2704, 2778, 3406, 6558, 6673, 6789, 7024, 9594, 9733, 10156, 11479, 19704, 19903, 20304, 20709, 20913, etc.
Conjecture: min_{1<=m<=n} T(n,m) <= T(n,M) for all M > n if n <> 5.
LINKS
Gabriel Mincu and Laurentiu Panaitopol, Properties of some functions connected to prime numbers, J. Inequal. Pure Appl. Math., 9 No. 1 (2008), Art. 12.
FORMULA
EXAMPLE
a(19) = 3 since 19 = 5*6/2 + 4, so the 19th term is T(6,4) = pi(6*4) - pi(6)*pi(4) = 9 - 3*2 = 3.
Triangular array begins:
n\ m 1 2 3 4 5 6 7 8 9 10 11 12 13 14
1 0
2 1 1
3 2 1 0
4 2 2 1 2
5 3 1 0 2 0
6 3 2 1 3 1 2
7 4 2 0 1 -1 1 -1
8 4 2 1 3 0 3 0 2
9 4 3 1 3 2 4 2 4 6
10 4 4 2 4 3 5 3 6 8 9
11 5 3 1 4 1 3 1 3 5 9 5
12 5 4 1 5 2 5 3 4 8 10 7 9
13 6 3 0 3 0 3 0 3 6 7 4 6 3
14 6 3 1 4 1 5 1 5 6 10 6 9 6 8
15 6 4 2 5 3 6 3 6 8 11 8 11 8 10 12
MATHEMATICA
t[n_, m_] := PrimePi[n*m] - PrimePi[n]*PrimePi[m]; Table[ t[n, m], {n, 13}, {m, n}] // Flatten
PROG
(PARI) T(n, m) = primepi(n*m) - primepi(n)*primepi(m);
tabl(nn) = for (n=1, nn, for (m=1, n, print1(T(n, m), ", ")); print); \\ Michel Marcus, Nov 08 2017
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Jonathan Sondow and Robert G. Wilson v, Nov 06 2017
STATUS
approved