A249368 - OEIS

%I #6 Nov 09 2014 00:12:34

%S 1,5,2,10,7,3,18,14,12,4,26,25,23,15,6,35,37,40,31,22,8,45,50,57,52,

%T 46,27,9,59,63,79,76,77,55,33,11,69,83,102,104,112,89,67,38,13,87,100,

%U 128,135,152,129,111,73,43,16,99,121,156,170,197,179,162,122

%N Rectangular array by antidiagonals:  t(n,k) is the position of prime(n)*k^2 when the numbers prime(j)*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 primes.  Then t(n,k) is the position of k*sqrt(prime(n)) 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    10   18   26    35    45

%e 2   7    14   25   37    50    63

%e 3   12   23   40   57    79    102

%e 4   15   31   52   76    104   135

%e 6   22   46   77   112   152   197

%e The numbers prime(1)*k^2 are (2,8,18,32,50,...);

%e the numbers prime(2)*k^2 are (3,12,27,48,75,...);

%e the numbers prime(3)*k^2 are (5,20,45,80,125,...);

%e the joint ranking of all such numbers is (2,3,5,7,8,...) = A229125, in which numbers of the form 2*k^2 occupy positions 1,5,10,17,... which is row 1 of the present array.  Similarly, the numbers 3*k^2 occupy positions 2,7,14,20,...

%t z = 20000; e[h_] := e[h] = Select[Range[2000], Prime[h]*(#^2) < z &];

%t t = Table[Prime[n]*e[n]^2, {n, 1, 2000}]; s = Sort[Flatten[t]];

%t u[n_, k_] := Position[s, Prime[n]*k^2];

%t TableForm[Table[u[n, k], {n, 1, 15}, {k, 1, 15}]]   (* A249368 array *)

%t Table[u[k, n - k + 1], {n, 15}, {k, 1, n}] // Flatten  (* A249368 sequence *)

%Y Cf. A229125, A249369, A249370, A000040.

%K nonn,tabl,easy

%O 1,2

%A _Clark Kimberling_, Oct 26 2014