A004053 -id:A004053 - cp's OEIS frontend

A063432 Triangle read by rows in which k-th entry in row n is representation of n in base k, for 1 <= k <= n.

1, 11, 10, 111, 11, 10, 1111, 100, 11, 10, 11111, 101, 12, 11, 10, 111111, 110, 20, 12, 11, 10, 1111111, 111, 21, 13, 12, 11, 10, 11111111, 1000, 22, 20, 13, 12, 11, 10, 111111111, 1001, 100, 21, 14, 13, 12, 11, 10, 1111111111, 1010, 101, 22, 20, 14, 13

Offset: 1

Henry Bottomley, Jul 20 2001

Representation of n in base 1 is defined to be a concatenation of n 1's.
It is difficult to write twenty-one in base 11 using decimal digits.
Representation in bases greater than 10 are written in base 10. This is really nasty! - N. J. A. Sloane, Dec 06 2002

			Rows start (1), (11, 10), (111, 11, 10), (1111, 100, 11, 10), etc.

Cf. A063431.
Rows are effectively the reverse of A001731, A001732, A001733, A001734, A001735, A001736, A008707, A008708, A008709, A008710, A008711, A008712, A008713, A008714, A008715, A008716, A008717, etc.
Columns are truncated versions of A000042, A007088, A007089, A007090, A007091, A007092, A007093, A007094, A007095, A000027 and perhaps A055649, etc.
Without the 1st column becomes A004053.

f[n_] := Flatten[ Append[ {FromDigits[ Table[1, {n}]] }, Table[ FromDigits[ IntegerDigits[n, i]], {i, 2, n}]]]; Flatten[ Table[ f[n], {n, 1, 10}]] (* Robert G. Wilson v *)

A327225 For any n >= 0, let u and v be such that 2 <= u < v and the digits of n in bases u and v are the same up to a permutation and v is minimized; a(n) = u.

Original entry on oeis.org

2, 2, 3, 4, 5, 6, 7, 3, 9, 4, 11, 5, 13, 4, 15, 7, 5, 5, 19, 6, 21, 5, 3, 7, 25, 6, 6, 13, 4, 9, 7, 7, 33, 8, 8, 11, 7, 7, 7, 19, 13, 13, 10, 10, 7, 7, 5, 9, 49, 9, 8, 5, 4, 10, 13, 13, 9, 9, 9, 19, 61, 10, 10, 10, 9, 9, 5, 9, 6, 13, 11, 11, 73, 10, 9, 12, 9

Offset: 0

Rémy Sigrist, Aug 27 2019

For any n >= 0, the sequence is well defined as the representation of n in any base b >= max(2, n+1) corresponds to a single digit n.

(n, u = A327225(n), v = A327226(n)) = (n, n+1, n+2) iff n = 1 or n is in A059809. - Bernard Schott, Aug 31 2019

			For n = 11:
- the representations of 11 in bases b = 2..9 are:
    b  11 in base b
    -  ------------
    2  "1011"
    3  "102"
    4  "23"
    5  "21"
    6  "15"
    7  "14"
    8  "13"
    9  "12"
- the representation in base 9 is the least that shows the same digits, up to order, to some former base, namely the base 5,
- hence a(11) = 5.

Rémy Sigrist, Table of n, a(n) for n = 0..10000

See A327226 for the corresponding v's.

Cf. A004053, A059809.

a(n) = { my (s=[]); for (v=2, oo, my (d=vecsort(digits(n,v))); if (setsearch(s,d), forstep (u=v-1, 2, -1, if (vecsort(digits(n,u))==d, return (u))), s=setunion(s,[d]))) }

a(n) <= max(2, n+1).

A347053 a(n) is the smallest base-10 number greater than 1 such that when written in all bases from base 2 to base n its leading digit is 1.

Original entry on oeis.org

2, 3, 4, 5, 78125, 67108864, 4747561509943, 4747561509943, 1123100968590805486067490785139871311149300510808491947285143687519677653346191462727898443913171541426176

Offset: 2

Scott R. Shannon, Aug 14 2021

Each term is a power of a number <= n. The terms given in the data section are a(2) = 2, a(3) = 3, a(4) = 4, a(5) = 5, a(6) = 5^7 = 78125, a(7) = 4^13 = 67108864, a(8) = a(9) = 7^15 = 4747561509943, and a(10) = 6^135 = 1123...6176 (106 digits). The other known terms (too large to write in the data section) are a(11) = a(12) = 10^421 (422 digits), a(13) = 8^2144 = 1678...1296 (1937 digits), and a(14) = a(15) = 7^7081 = 1377...6007 (5985 digits).
Assuming a(16) exists it is greater than 10^21000.
a(16) = a(17) = 6^132847 = 1145...3136 (103376 digits); a(18) = 18^521808 = 1719...0576 (655012 digits); a(19) = a(20) = 7^1192509 = 1043...3607 (1007788 digits). - Jon E. Schoenfield, Aug 17 2021
a(21) = 3^6959688 (3320616 digits). - Scott R. Shannon and Jon E. Schoenfield_, Aug 20 2021

			a(2) = 2 as 2 = 10_2, which has 1 as its leading digit.
a(3) = 3 as 3 = 11_2 = 10_3, each of which has 1 as its leading digit.
a(4) = 4 as 4 = 100_2 = 11_3 = 10_4, each of which has 1 as its leading digit.
a(5) = 5 as 5 = 101_2 = 12_3 = 11_4 = 10_5, each of which has 1 as its leading digit.
a(6) = 78125 as 78125 = 10011000100101101_2 = 10222011112_3 = 103010231_4 = 10000000_5 = 1401405_6, each of which has 1 as its leading digit.
a(7) = 67108864 as 67108864 = 100000000000000000000000000_2 = 11200021111001111_3 = 10000000000000_4 = 114134440424_5 = 10354213104_6 = 1443262444_7, each of which has 1 as its leading digit.

Cf. A004053, A258107, A181929.

I:=IntegerToString; N:=17; k:=2; e:=[]; P:=[]; for b in [2..N] do e[b]:=0; P[b]:=1; end for; B:=2; for n in [2..N] do OK:=false; while not OK do OK:=true; for b in [n..2 by -1] do if k ge P[b]*b then j:=Ilog(b, k div P[b]); if j gt 0 then e[b]+:=j; P[b]*:=b^j; end if; end if; if k div P[b] ne 1 then OK:=false; e[b]+:=1; P[b]*:=b; k:=P[b]; B:=b; break; end if; end for; end while; if IsPower(B) then , r:=IsPower(B); E:=Ilog(r,k); else r:=B; E:=e[B]; end if; "a("*I(n)* ") = "*I(r)*"^"*I(E); end for; // _Jon E. Schoenfield, Aug 19 2021

Table[r=Range[2,n];k=1;While[FreeQ[w=Union[First/@IntegerDigits[#^k,r]]&/@r,{1}],k++];(First@@Position[w,{1}]+1)^k,{n,2,10}] (* Giorgos Kalogeropoulos, Aug 17 2021 *)

A362842 a(1) = 1; a(2) = 2; for n > 2, a(n) is the smallest positive number that has not yet appeared that shares a factor with a(n-1) when both a(n-1) and a(n) are read as numbers in bases from one more than the maximum digit in a(n-1) and a(n), up to base 10.

Original entry on oeis.org

1, 2, 4, 6, 3, 9, 12, 24, 8, 20, 10, 30, 33, 11, 22, 26, 13, 39, 15, 48, 28, 14, 49, 7, 70, 16, 38, 19, 57, 69, 18, 56, 76, 36, 60, 40, 42, 21, 63, 66, 44, 46, 23, 92, 32, 64, 62, 31, 93, 27, 90, 5, 50, 55, 77, 84, 35, 80, 68, 17, 119, 34, 94, 47, 329, 91, 52, 96, 45, 95, 25, 190, 54, 98, 58, 29

Offset: 1

Scott R. Shannon, May 05 2023

This is a base variation of the EKG sequence A064413. Despite numbers with larger digits having to share a factor with a(n-1) in fewer bases than those with only small digits, and would therefore seemingly appear more frequently, the frequency of the digits 8 and 9, for example, in the first 200000 terms is the same as the smaller digits 0 to 7, so surprisingly this does not appear to influence the determination of a(n).

In the first 200000 terms the smallest unused number is 25411, which implies all numbers will eventually appear. In the same range the fixed points are 1, 2, 424, 507, 1261, 1577, 2461, 4311; it is likely no more appear.

			a(7) = 12 as the maximum digit in a(6) = 9 and 12 is 9, so a(6) and a(7) are only read as base 10 numbers, and 12 is the smallest unused number which shares a factor with 9 in base 10.
a(8) = 24 as the maximum digit in a(7) = 12 and 24 is 4, and 12_k shares a factor with 24_k when they are read as numbers in all bases k = 5,6,7,8,9,10. No unused smaller number has this property, e.g. a(8) cannot equal 8 as a(7) in base 9 is 12_9 = 11, which does not share a factor with 8_9 = 8. This is the first term to differ from A064413.
a(9) = 8 as the maximum digit in a(8) = 24 and 8 is 8, and 24_k shares a factor with 8_k when they are read as numbers in all bases k = 9,10.

Scott R. Shannon, Table of n, a(n) for n = 1..10000
Scott R. Shannon, Image of the first 100000 terms. The green line is a(n) = n.

Cf. A064413, A004053, A354087, A352763, A348086, A337687.

A359183 a(n) is the smallest number such that when written in all bases from base 2 to base n its leading digit equals the base - 1.

Original entry on oeis.org

1, 2, 54, 13122, 15258789062500

Offset: 2

Scott R. Shannon, Dec 18 2022

Each term can be represented in some base < n as a number < n multiplied by the base to some power. The terms given in the data section are a(2) = 1, a(3) = 2, a(4) = 54 = 2*3^3, a(5) = 13122 = 2*3^8, a(6) = 15258789062500 = 4*5^18, a(7) = 8158...4608 (186 digits) = 3*4^308. The other known terms (too large to write in the data section) are a(8) = 9532...8658 (3448 digits) = 2*3^7226, a(9) = a(10) = 9123...2500 (10344 digits) = 4*5^14798.

Assuming a(11) exists, it is greater than 10^22500.

			a(2) = 1 as 1 = 1_2, which has 1 = 2 - 1 as its leading digit.
a(3) = 2 as 2 = 10_2 = 2_3, which have 1 = 2 - 1 and 2 = 3 - 1 as their leading digits.
a(4) = 54 as 54 = 110110_2 = 2000_3 = 312_4, which have 1 = 2 - 1, 2 = 3 - 1 and 3 = 4 - 1 as their leading digits.
a(5) = 13122 as 13122 = 11001101000010_2 = 200000000_3 = 3031002_4 = 404442_5, which have 1 = 2 - 1, 2 = 3 - 1, 3 = 4 - 1 and 4 = 5 - 1 as their leading digits.
a(6) = 15258789062500 as 15258789062500 = 110000010110110101100111010011101100100_2 = 2000000201121020121212112011_3 = 3132002312230322131210_4 = 4000000000000000000_5 = 52241442501204004_6, which have 1 = 2 - 1, 2 = 3 - 1, 3 = 4 - 1, 4 = 5 - 1 and 5 = 6 - 1 as their leading digits.
a(7) = 81582795696655426358720748526459181157825502882872103403434619627581986794626\
  90448473536034793921827874140100908746255557234586263455831973302268738547817\
  2585724832003163984432734404608 (Too large to include in the DATA section)

Cf. A347053, A004053, A258107, A181929, A307254, A307255.

from math import floor, log
def a(n):
    arr = []
    p = 0
    while True:
        for m in range(1, n):
            for b in range(2, max(3, n)):
                    k = m*b**p
                    if k in arr:
                        continue
                    arr.append(k)
                    q = 1
                    for b in range(3, n+1):
                        if floor(k/b**floor(log(k)/log(b))) != b-1:
                            q = 0
                            break
                    if q:
                        return k
        p += 1
# Christoph B. Kassir, Feb 10 2023

cp's OEIS Frontend

A063432 Triangle read by rows in which k-th entry in row n is representation of n in base k, for 1 <= k <= n.

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A327225 For any n >= 0, let u and v be such that 2 <= u < v and the digits of n in bases u and v are the same up to a permutation and v is minimized; a(n) = u.

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A347053 a(n) is the smallest base-10 number greater than 1 such that when written in all bases from base 2 to base n its leading digit is 1.

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A362842 a(1) = 1; a(2) = 2; for n > 2, a(n) is the smallest positive number that has not yet appeared that shares a factor with a(n-1) when both a(n-1) and a(n) are read as numbers in bases from one more than the maximum digit in a(n-1) and a(n), up to base 10.

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A359183 a(n) is the smallest number such that when written in all bases from base 2 to base n its leading digit equals the base - 1.

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