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

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, 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, 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

Offset: 2

Author

Henry Bottomley, Nov 01 2002

Keywords

• base
• nonn
• tabl

Examples

			Rows start:
  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;
  2,2,0,0,0,0,0,1;
  etc.
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.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 2..1000

Crossrefs

Columns include A023416 and A077267. Row sums are A033093, row maxima are A062842, number of positive terms in each row are A077268.

Programs

Mathematica

Table[Count[#,0]&/@IntegerDigits[n,Range[2,n]],{n,2,15}]//Flatten (* Harvey P. Dale, Jun 02 2025 *)

PARI

T(n, k) = #select(x->(x==0), digits(n, k));
row(n) = vector(n-1, k, T(n,k+1));
tabl(nn) = for (n=2, nn, print(row(n))); \\ Michel Marcus, Sep 02 2020

Formula

T(nk, k)=T(n, k)+1; T(nk+m, k)=T(n, k) if 0

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