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%I #9 Nov 04 2014 11:13:33
%S 1,5,2,11,9,3,21,19,12,4,31,34,26,18,6,43,50,45,39,22,7,55,70,67,68,
%T 48,28,8,74,91,93,101,79,59,32,10,89,116,122,138,117,100,64,37,13,109,
%U 142,156,181,164,148,110,78,47,14,128,172,189,233,211,205,165
%N Rectangular array by descending antidiagonals: t(n,k) is the position of prime(n+1)*k^2 when the numbers prime(j+1)*h^2 are jointly ordered, for j>=1 and h>=1.
%C Equivalently, let S be the set of positive integer multiples of the square roots of the odd primes. Then t(n,k) is the position of k*sqrt(prime(n+1)) in the ordered union of S.
%C Every positive integer occurs exactly once in the array {t(n,k)}.
%e Northwest corner:
%e 1 5 11 21 31 43 55
%e 2 9 19 34 50 70 91
%e 3 12 26 45 67 93 122
%e 4 18 39 68 101 138 181
%e 6 22 48 79 117 164 211
%e The numbers 3*k^2 are (3,12,27,48,75,...);
%e the numbers 5*k^2 are (5,20,45,80,125,...);
%e the numbers 7*k^2 are (7,28,63,112,175,...);
%e the joint ranking of all such numbers is (3,5,7,11,12,13,...) = A249370,
%e in which numbers of the form 3*k^2 occupy positions 1,5,11,21,... which is row 1 of the present array. Similarly, the numbers 5*k^2 occupy positions 2,9,19,34,...
%t z = 20000; e[h_] := e[h] = Select[Range[2000], Prime[h]*(#^2) < z &];
%t t = Table[Prime[n]*e[n]^2, {n, 2, 2000}]; s = Sort[Flatten[t]];
%t u[n_, k_] := Position[s, Prime[n]*k^2];
%t TableForm[Table[u[n, k], {n, 2, 15}, {k, 1, 15}]] (* A249369 array *)
%t Table[u[k, n - k + 1], {n, 15}, {k, 1, n}] // Flatten (* A249369 sequence *)
%Y Cf. A249368, A249370.
%K nonn,tabl,easy
%O 1,2
%A _Clark Kimberling_, Oct 26 2014
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