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A286563 Triangular table T(n,k) read by rows: T(n,1) = 1, and for 1 < k <= n, T(n,k) = the highest exponent e such that k^e divides n.

%I A286563 #31 Apr 07 2025 10:42:19
%S A286563 1,1,1,1,0,1,1,2,0,1,1,0,0,0,1,1,1,1,0,0,1,1,0,0,0,0,0,1,1,3,0,1,0,0,
%T A286563 0,1,1,0,2,0,0,0,0,0,1,1,1,0,0,1,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,1,1,2,
%U A286563 1,1,0,1,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,0,0,0,1,0,0,0,0,0,0,1,1,0,1,0,1,0,0,0,0,0,0,0,0,0,1
%N A286563 Triangular table T(n,k) read by rows: T(n,1) = 1, and for 1 < k <= n, T(n,k) = the highest exponent e such that k^e divides n.
%C A286563 T(n,k) > 0 for k in row n of A027750. - _Michael De Vlieger_, May 20 2017
%C A286563 Compare rows to those of triangle A279907, smallest exponent e of n divisible by k. The values of k > -1 in row n of A279907 pertain to k in row n of A162306 rather than k in row n of A027750. - _Michael De Vlieger_, May 21 2017
%H A286563 Antti Karttunen, <a href="/A286563/b286563.txt">Table of n, a(n) for n = 1..10585; the first 145 rows of the triangle</a>
%F A286563 T(n,k) = A286561(n,k) listed row by row for n >= 1, k = 1 .. n.
%e A286563 The first fifteen rows of this triangular table:
%e A286563   1,
%e A286563   1, 1,
%e A286563   1, 0, 1,
%e A286563   1, 2, 0, 1,
%e A286563   1, 0, 0, 0, 1,
%e A286563   1, 1, 1, 0, 0, 1,
%e A286563   1, 0, 0, 0, 0, 0, 1,
%e A286563   1, 3, 0, 1, 0, 0, 0, 1,
%e A286563   1, 0, 2, 0, 0, 0, 0, 0, 1,
%e A286563   1, 1, 0, 0, 1, 0, 0, 0, 0, 1,
%e A286563   1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,
%e A286563   1, 2, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1,
%e A286563   1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,
%e A286563   1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1,
%e A286563   1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1
%p A286563 T := (n, k) -> ifelse(k = 1, 1, padic:-ordp(n, k)):
%p A286563 for n from 1 to 12 do seq(T(n, k), k = 1..n) od;  # _Peter Luschny_, Apr 07 2025
%t A286563 Table[If[k == 1, 1, IntegerExponent[n, k]], {n, 15}, {k, n}] // Flatten (* _Michael De Vlieger_, May 20 2017 *)
%o A286563 (Scheme) (define (A286563 n) (A286561bi (A002024 n) (A002260 n))) ;; For A286561bi see A286561.
%o A286563 (Python)
%o A286563 def T(n, k):
%o A286563     i=1
%o A286563     if k==1: return 1
%o A286563     while n%(k**i)==0:
%o A286563         i+=1
%o A286563     return i-1
%o A286563 for n in range(1, 21): print([T(n, k) for k in range(1, n + 1)]) # _Indranil Ghosh_, May 20 2017
%Y A286563 Lower triangular region of A286561.
%Y A286563 Cf. A286564 (same triangle reversed).
%Y A286563 Cf. A169594 (row sums).
%Y A286563 Cf. also arrays A051731, A286158, A027750, A279907, A280269.
%K A286563 nonn,tabl
%O A286563 1,8
%A A286563 _Antti Karttunen_, May 20 2017