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%I #15 Nov 08 2017 11:19:32
%S 0,1,1,2,1,0,2,2,1,2,3,1,0,2,0,3,2,1,3,1,2,4,2,0,1,-1,1,-1,4,2,1,3,0,
%T 3,0,2,4,3,1,3,2,4,2,4,6,4,4,2,4,3,5,3,6,8,9,5,3,1,4,1,3,1,3,5,9,5,5,
%U 4,1,5,2,5,3,4,8,10,7,9,6,3,0,3,0,3,0,3,6,7,4,6,3,6,3,1,4,1,5,1,5,6,10,6,9,6,8
%N Regular triangular array read by rows: T(n,m) = pi(n*m) - pi(n)*pi(m) for n > 0 and 0 < m <= n.
%C Inspired by A291440.
%C 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.
%C a(i) = -1 for i = 26 and 28, when n = 7 and m = either 5 or 7.
%C 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.
%C 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.
%C 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.
%C Conjecture: min_{1<=m<=n} T(n,m) <= T(n,M) for all M > n if n <> 5.
%H Gabriel Mincu and Laurentiu Panaitopol, <a href="https://www.emis.de/journals/JIPAM/article951.html">Properties of some functions connected to prime numbers</a>, J. Inequal. Pure Appl. Math., 9 No. 1 (2008), Art. 12.
%F a(n*(n+1)/2) = T(n,n) = A291440(n).
%F min_{1<=m<=n} a(n*(n-1)/2 + m) = min_{1<=m<=n} T(n,m) = A294509(n).
%e 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.
%e Triangular array begins:
%e n\ m 1 2 3 4 5 6 7 8 9 10 11 12 13 14
%e 1 0
%e 2 1 1
%e 3 2 1 0
%e 4 2 2 1 2
%e 5 3 1 0 2 0
%e 6 3 2 1 3 1 2
%e 7 4 2 0 1 -1 1 -1
%e 8 4 2 1 3 0 3 0 2
%e 9 4 3 1 3 2 4 2 4 6
%e 10 4 4 2 4 3 5 3 6 8 9
%e 11 5 3 1 4 1 3 1 3 5 9 5
%e 12 5 4 1 5 2 5 3 4 8 10 7 9
%e 13 6 3 0 3 0 3 0 3 6 7 4 6 3
%e 14 6 3 1 4 1 5 1 5 6 10 6 9 6 8
%e 15 6 4 2 5 3 6 3 6 8 11 8 11 8 10 12
%t t[n_, m_] := PrimePi[n*m] - PrimePi[n]*PrimePi[m]; Table[ t[n, m], {n, 13}, {m, n}] // Flatten
%o (PARI) T(n,m) = primepi(n*m) - primepi(n)*primepi(m);
%o tabl(nn) = for (n=1, nn, for (m=1, n, print1(T(n,m), ", ")); print); \\ _Michel Marcus_, Nov 08 2017
%Y Cf. A000720, A291440, A294509.
%K sign,tabl
%O 1,4
%A _Jonathan Sondow_ and _Robert G. Wilson v_, Nov 06 2017
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