cp's OEIS Frontend

A249369 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.

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