A077268 -id:A077268 - cp's OEIS frontend

A337496 Number of bases b for which the expansion of n in base b contains the largest digit possible (i.e., the digit b-1).

0, 1, 2, 2, 2, 3, 3, 4, 3, 3, 2, 5, 3, 4, 5, 5, 3, 5, 3, 6, 5, 5, 4, 8, 4, 4, 4, 5, 3, 8, 4, 6, 5, 5, 6, 8, 2, 3, 4, 7, 2, 7, 4, 7, 8, 7, 6, 11, 6, 7, 5, 6, 4, 8, 6, 8, 6, 6, 5, 12, 5, 6, 8, 7, 5, 7, 4, 7, 5, 9, 5, 12, 5, 6, 7, 7, 7, 9, 5, 11, 5, 3, 2, 11, 4, 3, 4, 8, 3, 11, 5

Offset: 0

François Marques, Aug 29 2020

An integer b > 1 is a main base of n if n in base b contains the largest digit possible (i.e., the digit b-1).
2 is a main base for all nonzero integers because in binary, they start with the digit 1.
10 is a main base for all numbers with a 9 in their decimal expansion.
b = n+1 is a main base of n when n > 0.
A000005(n+1) - 1 <= a(n) <= ceiling(sqrt(n+1)) + floor(A000005(n+1)/2) - 1 for n > 0. Indeed, if b != 1 is a divisor of n+1 then n = (k-1)*b + b-1 so the last digit of n in base b is b-1. On the other side, n = q*b + r with r < b. So if b is a main base of n then either r = b-1 (and b is a divisor of n+1) or b is a main base of q and therefore b-1 <= q which implies (b-1)^2 < n (i.e., b <= floor(sqrt(n))+1 <= ceiling(sqrt(n+1)) ). But if b > sqrt(n+1) is a divisor of (n+1) then (n+1)/b < sqrt(n+1) is another divisors of n+1 and only half of them can be greater than its square root. - François Marques, Dec 07 2020

			For n = 7, a(7) = 4 because the main bases of 7 are 2, 3, 4 and 8 as shown in the table below:
          Base b |   2 |   3 |   4 |   5 |   6 |   7 |   8
-----------------+-----+-----+-----+-----+-----+-----+-----
     7 in base b | 111 |  21 |  13 |  12 |  11 |  10 |   7
-----------------+-----+-----+-----+-----+-----+-----+-----
b is a main base | yes | yes | yes |  no |  no |  no | yes

François Marques, Table of n, a(n) for n = 0..10000
Devansh Singh, Link for Python Program below with comments

Cf. A077268 (contains digit 0).

A337496 := proc(n)
local k, r:=0;
for k from 2 to n+1 do
   if max(convert(n, base, k)) = k - 1 then
      r++;
   end if;
end do;
return r;
end proc:
seq(A337496(n), n=0..90);

baseQ[n_, b_] := MemberQ[IntegerDigits[n, b], b - 1]; a[n_] := Count[Range[2, n + 1], ?(baseQ[n, #] &)]; Array[a, 100, 0] (* _Amiram Eldar, Sep 01 2020 *)

a(n) = sum(b=2, n+1, vecmax(digits(n, b)) == b-1); \\ Michel Marcus, Aug 30 2020

a337496(n) = my(last_pos(v,k) = forstep(j=#v, 1, -1, if(v[j]==k, return(#v-j))); return(-1);, s=ceil(sqrt(n+1)), p); (n==0) + 1 + sum(b=2, s, p=last_pos(digits(n,b), b-1); if(p<0,0, p==0,2, 1)) -((n+1)==s^2) -2*((n+1)==s*(s-1)); \\ François Marques, Dec 07 2020

def A337496(N):
    A337496_n=[0,1]
    for j in range(2,N+1):
        A337496_n.append(2)
    for b in range(3,((N+1)//2) +1):
        n=2*b-1
        while n<=N:
            s=0
            m=n//b
            while m%b==b-2:
                s=s+1
                m=m//b
            x=b*((b**s)-1)//(b-1)
            for i in range(n, min(N,x+n)+1):
                A337496_n[i]+=1
            n=n+x+b
    return(A337496_n)
print(A337496(100)) # Devansh Singh, Dec 30 2020

a(n) <= (n+1)/2 for n >= 3. - Devansh Singh, Sep 21 2020

Minor edits by M. F. Hasler, Oct 26 2020

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

Original entry on oeis.org

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

Henry Bottomley, Nov 01 2002

			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.

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

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

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

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

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

A384436 a(n) is the number of distinct ways to represent n in any integer base >= 2 using only square digits.

Original entry on oeis.org

1, 1, 1, 2, 4, 3, 3, 3, 3, 6, 5, 4, 5, 5, 4, 4, 6, 5, 4, 5, 7, 7, 5, 5, 7, 8, 6, 6, 8, 7, 7, 7, 7, 7, 7, 6, 11, 9, 6, 7, 10, 7, 7, 7, 8, 8, 8, 6, 8, 11, 7, 7, 9, 10, 7, 7, 10, 10, 7, 7, 11, 10, 7, 7, 13, 11, 7, 7, 11, 10, 7, 7, 10, 11, 8, 8, 11, 11, 9, 8, 11, 15

Offset: 0

Felix Huber, May 29 2025

The representations of n remain the same for bases greater than n, as they all consist solely of the digit n.

			The a(36) = 11 distinct ways to represent 36 using only square digits are [1,0,0,1,0,0] in base 2, [1,1,0,0] in base 3, [1,0,0] in base 6, [4,4] in base 8, [4,0] in base 9, [1,16] in base 20, [1,9] in base 27, [1,4] in base 32, [1,1] in base 35, [1,0] in base 36 and [36] in bases >= 37.

Felix Huber, Table of n, a(n) for n = 0..10000

Cf. A046030, A055240, A061845, A077268, A126071, A135551, A384211, A384212.

A384436:=proc(n)
    local a,b,c;
    a:=0;
    for b from 2 to n+1 do
        c:=convert(n,'base',b);
        if select(issqr,c)=c then
            a:=a+1
        fi
    od;
    return max(1,a)
end proc;
seq(A384436(n),n=0..81);

a[n_] := Sum[Boole[AllTrue[IntegerDigits[n, b], IntegerQ[Sqrt[#]] &]], {b, 2, n+1}]; a[0] = 1; Array[a, 100, 0] (* Amiram Eldar, May 29 2025 *)

a(n) = sum(b=2, n+1, my(d=digits(n,b)); #select(issquare, d) == #d); \\ Michel Marcus, May 29 2025

Trivial lower bound for n >= 2: a(n) >= 2 for nonsquares n and a(n) >= 3 for squares n because in base 2 the representations of n consists only of the square digits '0' and '1', in base n the representation of n is [1,0] and in bases > n the representation of n is [n].

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cp's OEIS Frontend

A337496 Number of bases b for which the expansion of n in base b contains the largest digit possible (i.e., the digit b-1).

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

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Mathematica

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A384436 a(n) is the number of distinct ways to represent n in any integer base >= 2 using only square digits.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula