A122587 -id:A122587 - cp's OEIS frontend

A339255 Leading digit of n in base 5.

1, 2, 3, 4, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3

Offset: 1

Kevin Ryde, Nov 28 2020

Kevin Ryde, Table of n, a(n) for n = 1..10000

Cf. A007091 (base 5), A073851 (partial sums).

In other bases: A122586, A122587, A339256, A000030, A099563, A276153.

IntegerDigits[#,5][[1]]&/@Range[100] (* Harvey P. Dale, Sep 04 2021 *)

PARI
```
a(n) = n\5^logint(n,5);
```

a(n) = floor(n / 5^floor(log_5(n))).

G.f.: (x + Sum_{k>=0} Sum_{d=2..4} (x^(d*5^k)-x^(5^(k+1))) )/(1-x).

A339256 Leading digit of n in base 6.

Original entry on oeis.org

1, 2, 3, 4, 5, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 1

Kevin Ryde, Nov 28 2020

Kevin Ryde, Table of n, a(n) for n = 1..10000

Cf. A007092 (base 6), A109804 (partial sums).

In other bases: A122586, A122587, A339255, A000030, A099563, A276153.

Table[IntegerDigits[n,6][[1]],{n,90}] (* Harvey P. Dale, Jul 19 2023 *)

PARI
```
a(n) = n\6^logint(n,6);
```

a(n) = floor(n / 6^floor(log_6(n))).

G.f.: (x + Sum_{k>=0} Sum_{d=2..5} (x^(d*6^k)-x^(6^(k+1))) )/(1-x).

A361946 If the base-4 expansion of n starts with the digit 1, then replace 2's by 3's and vice versa; if it starts with the digit 2, then replace 1's by 3's and vice versa; if it starts with the digit 3, then replace 1's by 2's and vice versa; a(0) = 0.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 6, 8, 11, 10, 9, 12, 14, 13, 15, 16, 17, 19, 18, 20, 21, 23, 22, 28, 29, 31, 30, 24, 25, 27, 26, 32, 35, 34, 33, 44, 47, 46, 45, 40, 43, 42, 41, 36, 39, 38, 37, 48, 50, 49, 51, 56, 58, 57, 59, 52, 54, 53, 55, 60, 62, 61, 63, 64, 65, 67, 66

Offset: 0

Rémy Sigrist, Apr 01 2023

This sequence is a self-inverse permutation of the nonnegative integers.

			For n = 539:
- the base-4 expansion of 539 is "20123",
- it starts with the digit 2, so we replace 1's by 3's and vice versa,
- so the base-4 expansion of a(539) is "20321", and a(539) = 569.

Cf. A000695, A048647, A057300, A122587, A163241, A361945, A361947.

a(n) = { my (q = digits(n, 4), m = if (#q, [ [0,1,3,2], [0,3,2,1], [0,2,1,3] ][q[1]], [0,1,2,3])); fromdigits(apply (d -> m[1+d], q), 4); }

a(n) = A163241(n) when A122587(n) = 1.
a(n) = A048647(n) when A122587(n) = 2.
a(n) = A057300(n) when A122587(n) = 3.
a(n) = n iff n = d * A000695(k) for some d in {1, 2, 3} and some k >= 0.

A273619 Table read by antidiagonals (n>1, k>0): A(n,k) = leading digit of k in base n.

Original entry on oeis.org

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

Offset: 2

Andrey Zabolotskiy, May 30 2016

This is a generalization of A000030.

The first occurrence of a number k in the sequence is given by A(k+1,k).

			First few rows of the array are:
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1...
1, 2, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2...
1, 2, 3, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 1, 1, 1...
1, 2, 3, 4, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3...
1, 2, 3, 4, 5, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3...
1, 2, 3, 4, 5, 6, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2...
1, 2, 3, 4, 5, 6, 7, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2...
Note that the initial row is row 2.
A(3,3) corresponds to row n=3 and column k=3, and k=3 is written as 10 in base n=3, and the leading digit of 10 is 1, so A(3,3)=1.
A(12,11) corresponds to row n=12 and column k=11, and 11 is written as B in base 12, and the leading and only digit of B is B which is number 11 in decimal, so A(12,11)=11.

Robert Israel, Table of n, a(n) for n = 2..10012 (first 142 antidiagonals, flattened)

Cf. A000030 (row 10), A122586 (row 3), A122587 (row 4).

Cf. A051777, A051778 (may be interpreted as arrays of last digits of k in base n).

A:= (n,k) -> floor(k/n^floor(log[n](k))):
seq(seq(A(n-k,k),k=1..n-2),n=2..20); # Robert Israel, May 31 2016

a[n_, k_] := First[IntegerDigits[k, n]];

T(n,k) = digits(k, n)[1];
tabl(10, 10, n, k, n++; T(n,k)); \\ Michel Marcus, Jun 12 2016

From Robert Israel, May 31 2016: (Start)
A(n,k) = floor(k/n^floor(log_n(k))).
A(n,k) = k if n > k.
A(n,k) = A(n, floor(k/n)) otherwise.
G.f. of row n, G_n(x), satisfies G_n(x) = (1-x^n)/(1-x)^2 - (1+(n-1)*x^n)/(1-x) + (1-x^n)*G_n(x^n)/(1-x). (End)

cp's OEIS Frontend

A339255 Leading digit of n in base 5.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A339256 Leading digit of n in base 6.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A361946 If the base-4 expansion of n starts with the digit 1, then replace 2's by 3's and vice versa; if it starts with the digit 2, then replace 1's by 3's and vice versa; if it starts with the digit 3, then replace 1's by 2's and vice versa; a(0) = 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A273619 Table read by antidiagonals (n>1, k>0): A(n,k) = leading digit of k in base n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula