A338090 -id:A338090 - cp's OEIS frontend

A337239 Numbers having at least one 7 in their representation in base 8.

7, 15, 23, 31, 39, 47, 55, 56, 57, 58, 59, 60, 61, 62, 63, 71, 79, 87, 95, 103, 111, 119, 120, 121, 122, 123, 124, 125, 126, 127, 135, 143, 151, 159, 167, 175, 183, 184, 185, 186, 187, 188, 189, 190, 191, 199, 207, 215, 223, 231, 239, 247, 248, 249, 250, 251, 252, 253, 254, 255

Offset: 1

François Marques, Sep 20 2020

Complementary sequence to A037474.

			54 is not in the sequence since it is 66_8 in base 8, but 55 is in the sequence since it is 67_8 in base 8.

François Marques, Table of n, a(n) for n = 1..10000

Cf. Numbers with at least one digit b-1 in base b : A074940 (b=3), A337250 (b=4), A337572 (b=5), A333656 (b=6), A337141 (b=7), this sequence (b=8), A338090 (b=9), A011539 (b=10), A095778 (b=11).

Cf. Numbers with no digit b-1 in base b: A005836 (b=3), A023717 (b=4), A020654 (b=5), A037465 (b=6), A020657 (b=7), A037474 (b=8), A037477 (b=9), A007095 (b=10), A171397 (b=11).

seq(`if`(numboccur(7, convert(n, base, 8))>0, n, NULL), n=0..100);

Select[ Range[ 0, 100 ], (Count[ IntegerDigits[ #, 8 ], 7 ]>0)& ]

isok(m) = #select(x->(x==7), digits(m, 8)) >= 1;

def A337239(n):
    def f(x):
        s = oct(x)[2:]
        l = s.find('7')
        if l >= 0:
            s = s[:l]+'6'*(len(s)-l)
        return n+int(s,7)
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Dec 04 2024

A043485 Numbers having one 8 in base 9.

Original entry on oeis.org

8, 17, 26, 35, 44, 53, 62, 71, 72, 73, 74, 75, 76, 77, 78, 79, 89, 98, 107, 116, 125, 134, 143, 152, 153, 154, 155, 156, 157, 158, 159, 160, 170, 179, 188, 197, 206, 215, 224, 233, 234, 235, 236, 237, 238, 239, 240, 241, 251, 260, 269, 278, 287, 296, 305, 314

Offset: 1

Clark Kimberling

Cf. A007095 (numbers in base 9), A338090.

Select[Range[300],DigitCount[#,9,8]==1&] (* Harvey P. Dale, Jun 26 2011 *)

isok(m) = #select(x->(x==8), digits(m, 9)) == 1; \\ Michel Marcus, Oct 13 2020

Name edited by Michel Marcus, Oct 13 2020

A190598 Maximal digit in base-9 expansion of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 1, 1, 2, 3, 4, 5, 6, 7, 8, 2, 2, 2, 3, 4, 5, 6, 7, 8, 3, 3, 3, 3, 4, 5, 6, 7, 8, 4, 4, 4, 4, 4, 5, 6, 7, 8, 5, 5, 5, 5, 5, 5, 6, 7, 8, 6, 6, 6, 6, 6, 6, 6, 7, 8, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 1, 1, 2, 3, 4, 5, 6, 7, 8, 1, 1, 2, 3, 4, 5, 6, 7, 8, 2, 2, 2, 3, 4, 5

Offset: 0

N. J. A. Sloane, May 13 2011

Cf. A007095 (base 9), A033046 (indices of 1's), A338090 (indices of 8's).

Cf. A054055 (maximal digit in decimal).

a[n_] := Max[IntegerDigits[n, 9]]; (* Matej Veselovac, Jul 23 2021 *)

a(n) = if (n, vecmax(digits(n, 9)), 0); \\ Michel Marcus, Jul 19 2020

from sympy.ntheory.digits import digits
def a(n): return max(digits(n, 9)[1:])
print([a(n) for n in range(105)]) # Michael S. Branicky, Jul 23 2021

From Matej Veselovac, Jul 23 2021: (Start)
a(n) = 1 iff n is in A033046.
a(n) = 8 iff n is in A338090. (End)

A333970 Irregular triangle read by rows where the n-th row lists the bases 2<=b<=n+1 where n in base b contains the digit b-1.

Original entry on oeis.org

2, 2, 3, 2, 4, 2, 5, 2, 3, 6, 2, 3, 7, 2, 3, 4, 8, 2, 3, 9, 2, 5, 10, 2, 11, 2, 3, 4, 6, 12, 2, 4, 13, 2, 4, 7, 14, 2, 3, 4, 5, 15, 2, 3, 4, 8, 16, 2, 3, 17, 2, 3, 6, 9, 18, 2, 3, 19, 2, 3, 4, 5, 10, 20, 2, 3, 5, 7, 21, 2, 3, 5, 11, 22, 2, 3, 5, 23, 2, 3, 4, 5, 6, 8, 12, 24

Offset: 1

Devansh Singh, Sep 03 2020

If a number n has base 'b' representation = (... (b-1) A(j-1) ...A(3) A(2) A(1) A(0)) contains digit b-1, where b = q*(k+1)/k, k>=1 , and Sum_{i>=0} ((A(i)(mod b-q))*((b-q)^i)) > 0 then there exists  n' < n such that that n' in base b-q = b' contains digit b'-1 at the same place as n in base b and 0 <= (A(i)-A'(i))/b' <= (k+1)-((A'(i)+1)/b') (A'(i) is digit of n' in base b')for all i>=0.*
This condition is necessary and sufficient.
Proof that Condition is Necessary:
Since b-1 = b-q+q-1 and b' = q/k (as b = q*(k+1)/k). Therefore (b-1) (mod b') = (b'+q-1) (mod b') = (q-1) (mod b') = b'-1 :-(1).
n in base 'b' representation = (... (b-1) A(j-1) ...A(3) A(2) A(1) A(0)).Then n = Sum_{i>=0} (A(i)*(b^i)) = Sum_{i>=0} (A(i)*((b-q+q)^i)).
n = Sum_{i>=0} (A(i)*(b'^i)) +
    Sum_{i>=1} (A(i)*(b^i - b'^i))
   = Sum_{i>=0} (A'(i)*(b'^i)) + Sum_{i>=0} ((A(i)-A'(i))* (b'^i)) + Sum_{i>=1} (A(i)*(b^i - b'^i)),
         where A'(i) = A(i) (mod b').
Now n-Sum_{i>=0} ((A(i)-A'(i))*(b'^i))
       - Sum_{i>=1} (A(i)*(b^i - b'^i))
       = Sum_{i>=0} (A'(i)*(b'^i)).
Since A'(j) = A(j) (mod b') =  (b-1) (mod b') = b'-1(due to equation (1) above and A(j) = b-1.
Hence there exists n'  = Sum_{i>=0} (A'(i)*(b'^i)) > 0 containing digit b'-1 in base b'.
Table of n/b with cell containing T(n, b) = (n', b') for q = b/2. n' = Sum_{i>=0} (A'(i)*(b'^i))
n/b|  4  |  6  |  8  |  10 | 12
3  |(1,2)|     |     |     |
4  |     |     |     |     |
5  |     |(2,3)|     |     |
6  |     |     |     |     |
7  |(3,2)|     |(3,4)|     |
8  |     |     |     |     |
9  |     |     |     |(4,5)|
10 |     |     |     |     |
11 |(1,2)|(5,3)|     |     |(5,6)
Example: For table n/b in comments containing (n',b') in its cells.
For n = 7:
In base b = 4, n = 13 :- q = b' = 4/2 = 2, and n' = (3 mod (2))*(2)^0 + (1 mod(2))*(2)^1 = 1+2 = 3.
In base b = 8, n = 7 :- q = b' = 8/2 = 4, and n' = (7 mod (4))*(4)^0 = 3.
There are no other bases b >= 4 except 4, 8 for n = 7.
(n, b) maps to (0, 1) if b is prime. Following this and comment in A337536 we can say that all of the terms of A337536 will map to (0, 1) only, except A337536(2).
For above (n, b) -> (n', b') one possible (n, b) pair for (n', b') is { Sum_{i>=0} ((A'(i)+b') *((2*b')^i)), 2*b'}.

			Triangle begins
  Row    Bases
  n=1:   2
  n=2:   2  3
  n=3:   2  4
  n=4:   2  5
  n=5:   2  3  6
  n=6:   2  3  7
  n=7:   2  3  4  8
  n=8:   2  3  9
  n=9:   2  5  10
  n=10:  2  11

Rémy Sigrist, Table of n, a(n) for n = 1..10027 (rows for n = 1..1014, flattened)

Cf. A337535 (second column), A338295 (penultimate column), A337496 (row widths), A337536 (width 2), A337143 (width 3).

Rows containing bases 3..11 respectively: A074940, A337250, A337572, A333656, A337141, A337239, A338090, A011539, A095778.

row(n) = {my(list = List()); for (b=2, n+1, if (vecmax(digits(n, b)) == b-1, listput(list, b));); Vec(list);} \\ Michel Marcus, Sep 11 2020

More terms from Michel Marcus, Sep 11 2020

cp's OEIS Frontend

A337239 Numbers having at least one 7 in their representation in base 8.

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A043485 Numbers having one 8 in base 9.

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A190598 Maximal digit in base-9 expansion of n.

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A333970 Irregular triangle read by rows where the n-th row lists the bases 2<=b<=n+1 where n in base b contains the digit b-1.

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