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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.

%I A285730 #19 Jun 12 2025 12:57:44
%S A285730 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,
%T A285730 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,
%U A285730 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
%N 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.
%C A285730 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
%H A285730 Antti Karttunen, <a href="/A285730/b285730.txt">Table of n, a(n) for n = 1..10440; the first 144 antidiagonals of array</a>
%F A285730 A(n,1) = A006530(n), for k > 1, A(n,k) = A006530(n) * A(n/A006530(n),k-1).
%e A285730 The top left 5x18 corner of the array:
%e A285730    1,  1,  1,  1,  1
%e A285730    2,  2,  2,  2,  2
%e A285730    3,  3,  3,  3,  3
%e A285730    2,  4,  4,  4,  4
%e A285730    5,  5,  5,  5,  5
%e A285730    3,  6,  6,  6,  6
%e A285730    7,  7,  7,  7,  7
%e A285730    2,  4,  8,  8,  8
%e A285730    3,  9,  9,  9,  9
%e A285730    5, 10, 10, 10, 10
%e A285730   11, 11, 11, 11, 11
%e A285730    3,  6, 12, 12, 12
%e A285730   13, 13, 13, 13, 13
%e A285730    7, 14, 14, 14, 14
%e A285730    5, 15, 15, 15, 15
%e A285730    2,  4,  8, 16, 16
%e A285730   17, 17, 17, 17, 17
%e A285730    3,  9, 18, 18, 18
%e A285730 For A(18,1) we take just the largest prime factor of 18 = 2*3*3, thus A(18,1) = 3.
%e A285730 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.
%e A285730 For A(18,3) we take the product of three largest prime factors of 18, thus A(18,2) = 3*3*2 = 18.
%t A285730 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 *)
%o A285730 (Scheme)
%o A285730 (define (A285730 n) (A285730bi (A002260 n) (A004736 n)))
%o A285730 (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))))))
%o A285730 ;; Alternatively, implemented with the given recurrence formula:
%o A285730 (define (A285730bi row col) (if (= 1 col) (A006530 row) (* (A006530 row) (A285730bi (A052126 row) (- col 1)))))
%o A285730 (Python)
%o A285730 from sympy import primefactors
%o A285730 def a006530(n): return 1 if n==1 else max(primefactors(n))
%o A285730 def A(n, k): return a006530(n) if k==1 else a006530(n)*A(n//a006530(n), k - 1)
%o A285730 for n in range(1, 21): print([A(k, n - k + 1) for k in range(1, n + 1)]) # _Indranil Ghosh_, Apr 28 2017
%Y A285730 Transpose: A285731.
%Y A285730 Cf. A006530 (the leftmost column).
%Y A285730 Cf. A001222, A052126.
%K A285730 nonn,tabl
%O A285730 1,3
%A A285730 _Antti Karttunen_, Apr 28 2017