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A286561 Square array A(n,k): A(n,1) = 1, and for k > 1, A(n,k) = the highest exponent e such that k^e divides n, read by descending antidiagonals as A(1,1), A(1,2), A(2,1), etc.

%I A286561 #32 Mar 10 2021 03:22:37
%S A286561 1,0,1,0,1,1,0,0,0,1,0,0,1,2,1,0,0,0,0,0,1,0,0,0,1,0,1,1,0,0,0,0,0,1,
%T A286561 0,1,0,0,0,0,1,0,0,3,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,1,2,1,1,0,0,
%U A286561 0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,2,1,0,0,0,0,0,0,0,0,0,1,0,1,0,1,0,0,0,0,0,0,0,1,0,0,0,1,0,1,1
%N A286561 Square array A(n,k): A(n,1) = 1, and for k > 1, A(n,k) = the highest exponent e such that k^e divides n, read by descending antidiagonals as A(1,1), A(1,2), A(2,1), etc.
%H A286561 Antti Karttunen, <a href="/A286561/b286561.txt">Table of n, a(n) for n = 1..10585; the first 145 antidiagonals of array</a>
%e A286561 The top left 18 X 18 corner of the array:
%e A286561   n \k 1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18
%e A286561      .-----------------------------------------------------
%e A286561    1 | 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
%e A286561    2 | 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
%e A286561    3 | 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
%e A286561    4 | 1, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
%e A286561    5 | 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
%e A286561    6 | 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
%e A286561    7 | 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
%e A286561    8 | 1, 3, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
%e A286561    9 | 1, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0
%e A286561   10 | 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0
%e A286561   11 | 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0
%e A286561   12 | 1, 2, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0
%e A286561   13 | 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0
%e A286561   14 | 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0
%e A286561   15 | 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0
%e A286561   16 | 1, 4, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0
%e A286561   17 | 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0
%e A286561   18 | 1, 1, 2, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1
%e A286561   ---------------------------------------------------------
%e A286561 A(18,2) = 1, because 2^1 divides 18, but 2^2 does not. A(18,3) = 2, because 3^2 divides 18 (but 3^3 does not). A(18,4) = 0, because 4^0 (= 1) divides 18, but 4^1 does not. A(18,18) = 1, because 18^1 divides 18, but 18^2 does not.
%e A286561 A(2,18) = 0, because 18^0 divides 2, but 18^1 does not.
%t A286561 Table[Function[m, If[k == 1, 1, IntegerExponent[m, k]]][n - k + 1], {n, 15}, {k, n}] // TableForm (* _Michael De Vlieger_, May 20 2017 *)
%o A286561 (Scheme)
%o A286561 (define (A286561 n) (A286561bi (A002260 n) (A004736 n)))
%o A286561 (define (A286561bi row col) (if (= 1 col) 1 (let loop ((i 1)) (if (not (zero? (modulo row (expt col i)))) (- i 1) (loop (+ 1 i))))))
%o A286561 (PARI) A286561(n,k) = if(1==k, 1, valuation(n, k)); \\ _Antti Karttunen_, May 27 2017
%o A286561 (Python)
%o A286561 def a(n, k):
%o A286561     i=1
%o A286561     if k==1: return 1
%o A286561     while n%(k**i)==0:
%o A286561         i+=1
%o A286561     return i-1
%o A286561 for n in range(1, 21): print([a(k, n - k + 1) for k in range(1, n + 1)]) # _Indranil Ghosh_, May 20 2017
%Y A286561 Cf. A286562 (transpose), A286563 (lower triangular region), A286564 (lower triangular region reversed).
%Y A286561 Cf. A169594 (row sums), also A168512, A178638, A186643.
%Y A286561 Cf. also array A286156.
%K A286561 nonn,tabl
%O A286561 1,14
%A A286561 _Antti Karttunen_, May 20 2017