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A273619 Table read by antidiagonals (n>1, k>0): A(n,k) = leading digit of k in base n. 1
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

2,5

COMMENTS

This is a generalization of A000030.

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

LINKS

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

FORMULA

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)

EXAMPLE

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.

MAPLE

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

MATHEMATICA

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

PROG

(PARI) T(n, k) = digits(k, n)[1];

tabl(10, 10, n, k, n++; T(n, k)); \\ Michel Marcus, Jun 12 2016

CROSSREFS

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).

Sequence in context: A073204 A248641 A282934 * A034003 A168508 A177994

Adjacent sequences:  A273616 A273617 A273618 * A273620 A273621 A273622

KEYWORD

tabl,base,easy,nonn

AUTHOR

Andrey Zabolotskiy, May 30 2016

STATUS

approved

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Last modified October 19 11:31 EDT 2018. Contains 316359 sequences. (Running on oeis4.)