A285321 Square array A(1,k) = A019565(k), A(n,k) = A065642(A(n-1,k)), read by descending antidiagonals.

2, 3, 4, 6, 9, 8, 5, 12, 27, 16, 10, 25, 18, 81, 32, 15, 20, 125, 24, 243, 64, 30, 45, 40, 625, 36, 729, 128, 7, 60, 75, 50, 3125, 48, 2187, 256, 14, 49, 90, 135, 80, 15625, 54, 6561, 512, 21, 28, 343, 120, 225, 100, 78125, 72, 19683, 1024

Offset

1,1

Comments

A permutation of the natural numbers > 1.

Otherwise like array A284311, but the columns come in different order.

Links

Antti Karttunen, Table of n, a(n) for n = 1..120; the first 15 antidiagonals of array

Formula

A(1,k) = A019565(k), A(n,k) = A065642(A(n-1,k)).

For all n >= 2: A(A008479(n), A087207(n)) = n.

Example

The top left 12x6 corner of the array:
   2,   3,  6,     5,  10,  15,  30,      7,  14,  21,  42,   35
   4,   9, 12,    25,  20,  45,  60,     49,  28,  63,  84,  175
   8,  27, 18,   125,  40,  75,  90,    343,  56, 147, 126,  245
  16,  81, 24,   625,  50, 135, 120,   2401,  98, 189, 168,  875
  32, 243, 36,  3125,  80, 225, 150,  16807, 112, 441, 252, 1225
  64, 729, 48, 15625, 100, 375, 180, 117649, 196, 567, 294, 1715

Mathematica

a065642[n_] := Module[{k}, If[n == 1, Return[1], k = n + 1; While[ EulerPhi[k]/k != EulerPhi[n]/n, k++]]; k];
A[1, k_] := Times @@ Prime[Flatten[Position[#, 1]]]&[Reverse[ IntegerDigits[k, 2]]];
A[n_ /; n > 1, k_] := A[n, k] = a065642[A[n - 1, k]];
Table[A[n - k + 1, k], {n, 1, 10}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Nov 17 2019 *)

Prog

(Scheme)
(define (A285321 n) (A285321bi (A002260 n) (A004736 n)))
(define (A285321bi row col) (if (= 1 row) (A019565 col) (A065642 (A285321bi (- row 1) col))))
(Python)
from operator import mul
from sympy import prime, primefactors
def a019565(n): return reduce(mul, (prime(i+1) for i, v in enumerate(bin(n)[:1:-1]) if v == '1')) if n > 0 else 1 # This function from Chai Wah Wu
def a007947(n): return 1 if n<2 else reduce(mul, primefactors(n))
def a065642(n):
    if n==1: return 1
    r=a007947(n)
    n = n + r
    while a007947(n)!=r:
        n+=r
    return n
def A(n, k): return a019565(k) if n==1 else a065642(A(n - 1, k))
for n in range(1, 11): print([A(k, n - k + 1) for k in range(1, n + 1)]) # Indranil Ghosh, Apr 18 2017

Crossrefs

Transpose: A285322.

Cf. A019565, A065642.

Cf. A008479 (index of the row where n is located), A087207 (of the column).

Cf. arrays A284311, A285325, also A285332.

Sequence in context: A207826 A035312 A056230 * A253561 A354960 A119919

Adjacent sequences: A285318 A285319 A285320 * A285322 A285323 A285324

Keyword

nonn,tabl

Author

Antti Karttunen, Apr 17 2017

Status

approved