A337239 -id:A337239 - cp's OEIS frontend

A337141 Numbers having at least one 6 in their representation in base 7.

6, 13, 20, 27, 34, 41, 42, 43, 44, 45, 46, 47, 48, 55, 62, 69, 76, 83, 90, 91, 92, 93, 94, 95, 96, 97, 104, 111, 118, 125, 132, 139, 140, 141, 142, 143, 144, 145, 146, 153, 160, 167, 174, 181, 188, 189, 190, 191, 192, 193, 194, 195, 202, 209, 216, 223, 230, 237, 238, 239, 240

Offset: 1

François Marques, Sep 20 2020

Complementary sequence to A020657.

			33 is not in the sequence since it is 45_7 in base 7, but 34 is in the sequence since it is 46_7 in base 7.

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), this sequence (b=7), A337239 (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(6, convert(n, base, 7))>0, n, NULL), n=0..100);

Select[ Range[ 0, 100 ], (Count[ IntegerDigits[ #, 7 ], 6 ]>0)& ]
Select[Range[300],DigitCount[#,7,6]>0&] (* Harvey P. Dale, Dec 23 2020 *)

isok(m) = #select(x->(x==6), digits(m, 7)) >= 1;

from gmpy2 import digits
def A337141(n):
    def f(x):
        l = (s:=digits(x,7)).find('6')
        if l >= 0: s = s[:l]+'5'*(len(s)-l)
        return n+int(s,6)
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Dec 04 2024

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

A337141 Numbers having at least one 6 in their representation in base 7.

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