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A273619 Table read by antidiagonals (n>1, k>0): A(n,k) = leading digit of k in base n.

%I A273619 #48 Jun 27 2016 12:12:40
%S A273619 1,1,1,1,2,1,1,2,1,1,1,2,3,1,1,1,2,3,1,1,1,1,2,3,4,1,2,1,1,2,3,4,1,1,
%T A273619 2,1,1,2,3,4,5,1,1,2,1,1,2,3,4,5,1,1,2,1,1,1,2,3,4,5,6,1,1,2,1,1,1,2,
%U A273619 3,4,5,6,1,1,1,2,1,1,1,2,3,4,5,6,7,1
%N A273619 Table read by antidiagonals (n>1, k>0): A(n,k) = leading digit of k in base n.
%C A273619 This is a generalization of A000030.
%C A273619 The first occurrence of a number k in the sequence is given by A(k+1,k).
%H A273619 Robert Israel, <a href="/A273619/b273619.txt">Table of n, a(n) for n = 2..10012</a> (first 142 antidiagonals, flattened)
%F A273619 From _Robert Israel_, May 31 2016: (Start)
%F A273619 A(n,k) = floor(k/n^floor(log_n(k))).
%F A273619 A(n,k) = k if n > k.
%F A273619 A(n,k) = A(n, floor(k/n)) otherwise.
%F A273619 G.f. of row n, G_n(x), satisfies G_n(x) = (1-x^n)/(1-x)^2 - (1+(n-1)*x^n)/(1-x) + (1-x^n)*G_n(x^n)/(1-x). (End)
%e A273619 First few rows of the array are:
%e A273619 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...
%e A273619 1, 2, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2...
%e A273619 1, 2, 3, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 1, 1, 1...
%e A273619 1, 2, 3, 4, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3...
%e A273619 1, 2, 3, 4, 5, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3...
%e A273619 1, 2, 3, 4, 5, 6, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2...
%e A273619 1, 2, 3, 4, 5, 6, 7, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2...
%e A273619 Note that the initial row is row 2.
%e A273619 A(3,3) corresponds to row n=3 and column k=3, and k=3 is written as 10 in base n=3, and the leading digit of 10 is 1, so A(3,3)=1.
%e A273619 A(12,11) corresponds to row n=12 and column k=11, and 11 is written as B in base 12, and the leading and only digit of B is B which is number 11 in decimal, so A(12,11)=11.
%p A273619 A:= (n,k) -> floor(k/n^floor(log[n](k))):
%p A273619 seq(seq(A(n-k,k),k=1..n-2),n=2..20); # _Robert Israel_, May 31 2016
%t A273619 a[n_, k_] := First[IntegerDigits[k, n]];
%o A273619 (PARI) T(n,k) = digits(k, n)[1];
%o A273619 tabl(10, 10, n, k, n++; T(n,k)); \\ _Michel Marcus_, Jun 12 2016
%Y A273619 Cf. A000030 (row 10), A122586 (row 3), A122587 (row 4).
%Y A273619 Cf. A051777, A051778 (may be interpreted as arrays of last digits of k in base n).
%K A273619 tabl,base,easy,nonn
%O A273619 2,5
%A A273619 _Andrey Zabolotskiy_, May 30 2016