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A077266 Triangle of number of zeros when n is written in base k (2 <= k <= n).

%I A077266 #20 Jun 02 2025 17:45:18
%S A077266 1,0,1,2,0,1,1,0,0,1,1,1,0,0,1,0,0,0,0,0,1,3,0,1,0,0,0,1,2,2,0,0,0,0,
%T A077266 0,1,2,1,0,1,0,0,0,0,1,1,1,0,0,0,0,0,0,0,1,2,1,1,0,1,0,0,0,0,0,1,1,0,
%U A077266 0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,1,0,0,0,0,0,0,1,0,1,0,1,0,0,0,0,0,0,0,0,0,1
%N A077266 Triangle of number of zeros when n is written in base k (2 <= k <= n).
%H A077266 Harvey P. Dale, <a href="/A077266/b077266.txt">Table of n, a(n) for n = 2..1000</a>
%F A077266 T(nk, k)=T(n, k)+1; T(nk+m, k)=T(n, k) if 0<m<k; T(k, k)=1; T(n, k)=0 if n/2<k<n or 0<n<k.
%e A077266 Rows start:
%e A077266   1;
%e A077266   0,1;
%e A077266   2,0,1;
%e A077266   1,0,0,1;
%e A077266   1,1,0,0,1;
%e A077266   0,0,0,0,0,1;
%e A077266   3,0,1,0,0,0,1;
%e A077266   2,2,0,0,0,0,0,1;
%e A077266   etc.
%e A077266 9 can be written in bases 2-9 as: 1001, 100, 21, 14, 13, 12, 11 and 10, in which case the numbers of zeros are 2,2,0,0,0,0,0,1.
%t A077266 Table[Count[#,0]&/@IntegerDigits[n,Range[2,n]],{n,2,15}]//Flatten (* _Harvey P. Dale_, Jun 02 2025 *)
%o A077266 (PARI) T(n, k) = #select(x->(x==0), digits(n, k));
%o A077266 row(n) = vector(n-1, k, T(n,k+1));
%o A077266 tabl(nn) = for (n=2, nn, print(row(n))); \\ _Michel Marcus_, Sep 02 2020
%Y A077266 Columns include A023416 and A077267. Row sums are A033093, row maxima are A062842, number of positive terms in each row are A077268.
%K A077266 base,nonn,tabl
%O A077266 2,4
%A A077266 _Henry Bottomley_, Nov 01 2002