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A285730 Square array: If A001222(n) < k, then A(n,k) = n, otherwise A(n,k) = product of k largest prime factors of n (taken with multiplicity), read by descending antidiagonals. 3
1, 1, 2, 1, 2, 3, 1, 2, 3, 2, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 3, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 2, 1, 2, 3, 4, 5, 6, 7, 4, 3, 1, 2, 3, 4, 5, 6, 7, 8, 9, 5, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 3, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 6, 13, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 7 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Square array A(n,k) [where n is row and k is column] is read by descending antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..10440; the first 144 antidiagonals of array

FORMULA

A(n,1) = A006530(n), for k > 1, A(n,k) = A006530(n) * A(n/A006530(n),k-1).

EXAMPLE

The top left 5x18 corner of the array:

   1,  1,  1,  1,  1

   2,  2,  2,  2,  2

   3,  3,  3,  3,  3

   2,  4,  4,  4,  4

   5,  5,  5,  5,  5

   3,  6,  6,  6,  6

   7,  7,  7,  7,  7

   2,  4,  8,  8,  8

   3,  9,  9,  9,  9

   5, 10, 10, 10, 10

  11, 11, 11, 11, 11

   3,  6, 12, 12, 12

  13, 13, 13, 13, 13

   7, 14, 14, 14, 14

   5, 15, 15, 15, 15

   2,  4,  8, 16, 16

  17, 17, 17, 17, 17

   3,  9, 18, 18, 18

For A(18,1) we take just the largest prime factor of 18 = 2*3*3, thus A(18,1) = 3.

For A(18,2) we take the product of two largest prime factors of 18 (duplicates not discarded), thus A(18,2) = 3*3 = 9.

For A(18,3) we take the product of three largest prime factors of 18, thus A(18,2) = 3*3*2 = 18.

MATHEMATICA

With[{nn = 14}, Function[s, Table[s[[#, k]] &[n - k + 1], {n, nn}, {k, n, 1, -1}]]@ MapIndexed[PadRight[#1, nn, First@ #2] &, Table[FoldList[Times, Reverse@ Flatten[FactorInteger[n] /. {p_, e_} /; e > 0 :> ConstantArray[p, e]]], {n, nn}]]] // Flatten (* Michael De Vlieger, Apr 28 2017 *)

PROG

(Scheme)

(define (A285730 n) (A285730bi (A002260 n) (A004736 n)))

(define (A285730bi row col) (let loop ((n row) (k col) (m 1)) (if (zero? k) m (loop (/ n (A006530 n)) (- k 1) (* m (A006530 n))))))

;; Alternatively, implemented with the given recurrence formula:

(define (A285730bi row col) (if (= 1 col) (A006530 row) (* (A006530 row) (A285730bi (A052126 row) (- col 1)))))

(Python)

from sympy import primefactors

def a006530(n): return 1 if n==1 else max(primefactors(n))

def A(n, k): return a006530(n) if k==1 else a006530(n)*A(n/a006530(n), k - 1)

for n in xrange(1, 21): print [A(k, n - k + 1) for k in xrange(1, n + 1)] # Indranil Ghosh, Apr 28 2017

CROSSREFS

Transpose: A285731.

Cf. A006530 (the leftmost column).

Cf. A001222, A052126.

Sequence in context: A194436 A061336 A057945 * A280055 A253092 A194546

Adjacent sequences:  A285727 A285728 A285729 * A285731 A285732 A285733

KEYWORD

nonn,tabl

AUTHOR

Antti Karttunen, Apr 28 2017

STATUS

approved

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Last modified September 19 04:39 EDT 2019. Contains 327187 sequences. (Running on oeis4.)