A284311 Array T(n,k) read by antidiagonals (downward): T(1,k) = A005117(k+1) (squarefree numbers > 1); for n > 1, columns are nonsquarefree numbers (in ascending order) with exactly the same prime factors as T(1,k).

2, 3, 4, 5, 9, 8, 6, 25, 27, 16, 7, 12, 125, 81, 32, 10, 49, 18, 625, 243, 64, 11, 20, 343, 24, 3125, 729, 128, 13, 121, 40, 2401, 36, 15625, 2187, 256, 14, 169, 1331, 50, 16807, 48, 78125, 6561, 512, 15, 28, 2197, 14641, 80, 117649, 54, 390625, 19683, 1024

Offset: 1

Author

Bob Selcoe, Mar 24 2017

Keywords

• nonn
• tabl

Comments

A permutation of the natural numbers > 1.

T(1,k)= A005117(m) with m > 1; terms in column k = T(1,k) * A162306(T(1,k)) only not bounded by T(1,k). Let T(1,k) = b. Terms in column k are multiples of b and numbers c such that c | b^e with e >= 0. Alternatively, terms in column k are multiples bc with c those numbers whose prime divisors p also divide b. - Michael De Vlieger, Mar 25 2017

Examples

			Array starts:
    2    3     5  6      7  10       11        13  14  15
    4    9    25 12     49  20      121       169  28  45
    8   27   125 18    343  40     1331      2197  56  75
   16   81   625 24   2401  50    14641    371293  98 135
   32  243  3125 36  16807  80   161051   4826809 112 225
   64  729 15625 48 117649 100  1771561  62748517 196 375
  128 2187 78125 54 823543 160 19487171 815730721 224 405
Column 6 is: T(1,6) = 2*5; T(2,6) = 2^2*5; T(3,6) = 2^3*5; T(4,6) = 2*5^2; T(5,6) = 2^4*5, etc.
		

Links

• Alois P. Heinz, Antidiagonals n = 1..141, flattened

Crossrefs

Cf. A005117 (squarefree numbers), A033845 (column 4), columns 1,2,3,5 are powers of primes, A033846 (column 6), A033847 (column 9), A033849 (column 10).

The columns that are powers of primes have indices A071403(n) - 1. - Michel Marcus, Mar 24 2017

See also A007947; the k-th column of the array corresponds to the numbers with radical A005117(k+1). - Rémy Sigrist, Mar 24 2017

Cf. A284457 (this sequence read by antidiagonals upwards), A285321 (a similar array, but columns come in different order).

Cf. A065642.

Cf. A008479 (index of the row where n is located), A285329 (of the column).

Programs

Mathematica

f[n_, k_: 1] := Block[{c = 0, sgn = Sign[k], sf}, sf = n + sgn; While[c < Abs[k], While[! SquareFreeQ@ sf, If[sgn < 0, sf--, sf++]]; If[sgn < 0, sf--, sf++]; c++]; sf + If[sgn < 0, 1, -1]] (* after Robert G. Wilson v at A005117 *); T[n_, k_] := T[n, k] = Which[And[n == 1, k == 1], 2, k == 1, f@ T[n - 1, k], PrimeQ@ T[n, 1], T[n, 1]^k, True, Module[{j = T[n, k - 1]/T[n, 1] + 1}, While[PowerMod[T[n, 1], j, j] != 0, j++]; j T[n, 1]]]; Table[T[n - k + 1, k], {n, 10}, {k, n}] // Flatten (* Michael De Vlieger, Mar 25 2017 *)

Scheme

(define (A284311 n) (A284311bi  (A002260 n) (A004736 n)))
(define (A284311bi row col) (if (= 1 row) (A005117 (+ 1 col)) (A065642 (A284311bi (- row 1) col))))
;; Antti Karttunen, Apr 17 2017

Formula

From Antti Karttunen, Apr 17 2017: (Start)

A(1,k) = A005117(1+k), A(n,k) = A065642(A(n-1,k)).

A(A008479(n), A285329(n)) = n for all n >= 2.

(End)