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Table T(n, k), n > 1, k = 1..n-1, read by rows; T(n, k) is the greatest m such that the points (m, prime(m)), (k, prime(k)) and (n, prime(n)) are aligned (where prime(k) denotes the k-th prime number).
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%I #6 Sep 30 2024 10:57:31

%S 2,3,4,4,4,4,5,8,9,23,6,6,6,8,6,7,7,9,7,9,21,8,8,8,8,8,8,8,9,9,9,9,9,

%T 9,9,15,10,10,10,10,15,21,21,52,152,11,11,11,11,11,11,11,15,15,11,12,

%U 12,12,12,12,21,21,12,12,21,153,13,13,13,13,13,21,21,28,17,21,53,21

%N Table T(n, k), n > 1, k = 1..n-1, read by rows; T(n, k) is the greatest m such that the points (m, prime(m)), (k, prime(k)) and (n, prime(n)) are aligned (where prime(k) denotes the k-th prime number).

%H Rémy Sigrist, <a href="/A376571/b376571.txt">Table of n, a(n) for n = 2..10012</a> (rows for n = 2..142 flattened)

%F T(n, k) >= n.

%e Table T(n, k) begins:

%e 2;

%e 3, 4;

%e 4, 4, 4;

%e 5, 8, 9, 23;

%e 6, 6, 6, 8, 6;

%e 7, 7, 9, 7, 9, 21;

%e 8, 8, 8, 8, 8, 8, 8;

%e 9, 9, 9, 9, 9, 9, 9, 15;

%e 10, 10, 10, 10, 15, 21, 21, 52, 152;

%e 11, 11, 11, 11, 11, 11, 11, 15, 15, 11;

%e 12, 12, 12, 12, 12, 21, 21, 12, 12, 21, 153;

%e 13, 13, 13, 13, 13, 21, 21, 28, 17, 21, 53, 21;

%e ...

%o (PARI) T(n,k) = { my (x0 = k, y0 = prime(x0), x1 = n, y1 = prime(x1), s = (y1-y0)/(x1-x0), maxp = max(60184, exp(max(y0/x0, s) + 1.1)), x2 = 0, v = -oo); forprime (y2 = 2, 1+maxp, x2++; if (x0 * (y1 - y2) + x1 * (y2 - y0) + x2 * (y0 - y1)==0, v = x2;);); return (v); }

%Y Cf. A376187, A376569, A376570.

%K nonn,tabl

%O 2,1

%A _Rémy Sigrist_, Sep 28 2024