A249368 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.

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, 46, 27, 9, 59, 63, 79, 76, 77, 55, 33, 11, 69, 83, 102, 104, 112, 89, 67, 38, 13, 87, 100, 128, 135, 152, 129, 111, 73, 43, 16, 99, 121, 156, 170, 197, 179, 162, 122

Offset: 1

Author

Clark Kimberling, Oct 26 2014

Keywords

• nonn
• tabl
• easy

Comments

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.

Every positive integer occurs exactly once in the array {t(n,k)}.

Examples

			Northwest corner:
1   5    10   18   26    35    45
2   7    14   25   37    50    63
3   12   23   40   57    79    102
4   15   31   52   76    104   135
6   22   46   77   112   152   197
The numbers prime(1)*k^2 are (2,8,18,32,50,...);
the numbers prime(2)*k^2 are (3,12,27,48,75,...);
the numbers prime(3)*k^2 are (5,20,45,80,125,...);
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,...
		

Crossrefs

Cf. A229125, A249369, A249370, A000040.

Programs

Mathematica

z = 20000; e[h_] := e[h] = Select[Range[2000], Prime[h]*(#^2) < z &];
t = Table[Prime[n]*e[n]^2, {n, 1, 2000}]; s = Sort[Flatten[t]];
u[n_, k_] := Position[s, Prime[n]*k^2];
TableForm[Table[u[n, k], {n, 1, 15}, {k, 1, 15}]]   (* A249368 array *)
Table[u[k, n - k + 1], {n, 15}, {k, 1, n}] // Flatten  (* A249368 sequence *)