A063945 -id:A063945 - cp's OEIS frontend

A216653 Number A(n,k) of n-digit k-th powers; square array A(n,k), n>=1, k>=1, read by antidiagonals.

10, 4, 90, 3, 6, 900, 2, 2, 22, 9000, 2, 2, 5, 68, 90000, 2, 1, 2, 12, 217, 900000, 2, 1, 1, 4, 25, 683, 9000000, 2, 0, 1, 3, 8, 53, 2163, 90000000, 2, 0, 1, 1, 3, 14, 116, 6837, 900000000, 2, 0, 1, 1, 2, 6, 25, 249, 21623, 9000000000

Offset: 1

Alois P. Heinz, Sep 12 2012

			A(1,1) = 10: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
A(1,2) = 4: 0, 1, 4, 9.
A(2,2) = 6: 16, 25, 36, 49, 64, 81.
A(3,3) = 5: 125, 216, 343, 512, 729.
A(4,4) = 4: 1296, 2401, 4096, 6561.
A(5,5) = 3: 16807, 32768, 59049.
A(6,6) = 3: 117649, 262144, 531441.
Square array A(n,k) begins:
:n\k|        1:     2:    3:   4:   5:  6:  7:  8
+---+--------------------------------------------
: 1 |       10,     4,    3,   2,   2,  2,  2,  2
: 2 |       90,     6,    2,   2,   1,  1,  0,  0
: 3 |      900,    22,    5,   2,   1,  1,  1,  1
: 4 |     9000,    68,   12,   4,   3,  1,  1,  1
: 5 |    90000,   217,   25,   8,   3,  2,  2,  1
: 6 |   900000,   683,   53,  14,   6,  3,  2,  1
: 7 |  9000000,  2163,  116,  25,  10,  5,  2,  2
: 8 | 90000000,  6837,  249,  43,  14,  7,  4,  2

Alois P. Heinz, Antidiagonals n = 1..141, flattened

Columns k=1-10 give: A063945, A062940, A062941, A102831, A216655, A216656, A216657, A216658, A216659, A216654.

Main diagonal gives: A102690.

r:= proc(n, k) local b; b:= iroot(n, k); b+`if`(b^k r(10^n, k) -r(10^(n-1), k) +`if`(n=1, 1, 0):
seq(seq(A(n, 1+d-n), n=1..d), d=1..10);

A178500 a(n) = 10^n * signum(n).

Original entry on oeis.org

0, 10, 100, 1000, 10000, 100000, 1000000, 10000000, 100000000, 1000000000, 10000000000, 100000000000, 1000000000000, 10000000000000, 100000000000000, 1000000000000000, 10000000000000000, 100000000000000000, 1000000000000000000, 10000000000000000000, 100000000000000000000

Offset: 0

Reinhard Zumkeller, May 28 2010

a(n-1) is the minimum difference between an n-digit number (written in base 10, nonzero leading digit) and the product of its digits. For n > 1, it is also a number meeting that bound. See A070565. - Devin Akman, Apr 17 2019

Michael De Vlieger, Table of n, a(n) for n = 0..999
Index entries for linear recurrences with constant coefficients, signature (10).

Cf. A000533, A011557, A057427, A061601, A070565, A109002, A155559, A178501.

Partial sums of A063945.

Array[10^#*Sign[#] &, 20, 0] (* Michael De Vlieger, Apr 21 2019 *)

a(n) = A011557(n)*A057427(n).
For n > 0, a(n) = A011557(n).
a(n) = 10*A178501(n).
a(n) = A000533(n) - 1.
A061601(a(n)) = A109002(n+1).
From Elmo R. Oliveira, Jul 21 2025: (Start)
G.f.: 10*x/(1-10*x).
E.g.f.: 2*exp(5*x)*sinh(5*x).
a(n) = 10*a(n-1) for n > 1. (End)

A378564 a(n) is the number of n-digit nonnegative integers with the median of the digits equal to one of the digits.

Original entry on oeis.org

10, 9, 900, 1665, 90000, 232710, 9000000, 29055165, 900000000, 3413319138, 90000000000, 386095933170, 9000000000000, 42568084276236, 900000000000000, 4607838122919165, 90000000000000000, 491998811785538730, 9000000000000000000, 51983526276872387430, 900000000000000000000, 5447302810160797285236

Offset: 1

Stefano Spezia, Dec 01 2024

			From _David A. Corneth_, Dec 03 2024: (Start)
a(3) = 900 as every positive integer between (inclusive) 100 and 999 contains its median. The median is the middle digit after sorting which is in the digits.
a(4) = 1665. For example 2558 has digits sorted and the median, 5 is in the digits of 2558 and any permutation of digits of 2558. There are 12 such permutations so 2558 contributes 12 towards the total of a(4).
0258 has digits sorted (but a leading 0) and has 24 permutations. To account for the leading 0 we remove it and deduce the number of permutations from what is left, namely 258. That has 6 permutations. So in total 0258 adds 24 - 6 = 18 towards the total of a(4). (End)

Chai Wah Wu, Table of n, a(n) for n = 1..47 (terms 1..27 from David A. Corneth)
David A. Corneth, PARI program

Cf. A063945, A179239, A378560.

a[n_]:=If[OddQ[n], KroneckerDelta[n,1]+9*10^(n-1), Module[{c=0}, For[k=10^(n-1), k<=10^n-1, k++, If[MemberQ[digits=IntegerDigits[k], Median[digits]], c++]]; c]]; Array[a, 7]

PARI
```
\\ See Corneth link
```

from math import prod, factorial
from itertools import combinations_with_replacement
from collections import Counter
def A378564(n):
    if n==1: return 10
    if n&1: return 9*10**(n-1)
    c, f = 0, factorial(n-1)
    for p in combinations_with_replacement(range(10),n):
        if max(p):
            a = sorted(p)
            b = a[len(a)-1>>1]+a[len(a)>>1]
            if b&1^1 and b>>1 in p:
                v = Counter(d for d in p if d).values()
                s = sum(v)
                q = prod((factorial(i) for i in v))*factorial(n-s)
                c += sum(f*i//q for i in v)
    return c # Chai Wah Wu, Dec 14 2024

a(2*n-1) = 9*10^(n-1) with a(1) = 10.

a(n) = A063945(n) for n odd.

More terms from David A. Corneth, Dec 03 2024

A353962 Square array read by descending antidiagonals: The n-th row gives the decimal expansion of the base-n Champernowne constant.

Original entry on oeis.org

8, 6, 5, 2, 9, 4, 2, 8, 2, 3, 4, 9, 6, 1, 2, 0, 5, 1, 0, 3, 1, 1, 8, 1, 7, 9, 9, 1, 2, 1, 1, 3, 8, 4, 6, 1, 5, 6, 1, 6, 6, 4, 3, 4, 1, 8, 7, 1, 1, 2, 3, 2, 0, 2, 1, 6, 5, 1, 1, 6, 5, 6, 6, 3, 0, 0, 8, 3, 1, 1, 8, 5, 4, 2, 4, 9, 9, 0, 0, 8, 1, 1, 5, 3, 8, 4, 5, 9, 9, 9, 0

Offset: 2

Davis Smith, May 12 2022

The base-n Champernowne constant (C_n) is normal in base n. A(n,k) is the (k+1)-th decimal digit of the fractional part of C_n.

			The square array A(n,k) begins:
  n/k | 0  1  2  3  4  5  6  7  8  9 10 11 ...
  ----+---------------------------------------
   2  | 8  6  2  2  4  0  1  2  5  8  6  8 ...
   3  | 5  9  8  9  5  8  1  6  7  5  3  8 ...
   4  | 4  2  6  1  1  1  1  1  1  1  1  1 ...
   5  | 3  1  0  7  3  6  1  1  1  1  1  1 ...
   6  | 2  3  9  8  6  2  6  8  5  8  1  5 ...
   7  | 1  9  4  4  3  5  5  3  5  0  8  6 ...
   8  | 1  6  3  2  6  4  8  1  2  1  0  5 ...
   9  | 1  4  0  6  2  4  9  7  6  1  1  9 ...
  10  | 1  2  3  4  5  6  7  8  9  1  0  1 ...
  ...

Verónica Becher and Santiago Figueira, An example of a computable absolutely normal number, Theoretical Computer Science, 270 (2002), 947-958.
Arthur H. Copeland and Paul Erdős, Note on normal numbers, Bull. Amer. Math. Soc. 52 (1946), 857-860.
Davar Khoshnevisan, Normal Numbers are Normal, Clay Mathematics Institute Annual Report, 2006, 15-31.
Ivan Niven and H.S. Zuckerman, On The Definition of Normal Numbers, Pacific J. Math., 1 (1951), 103-109.
Davis Smith, A Sufficient Condition For Normalcy.

Rows: A066716 (n=2), A077771 (n=3), A033307 (n=10).

Cf. A063945.

A[n_,k_]:=Mod[Floor[ChampernowneNumber[n]10^(k + 1)] ,10]; Flatten[Table[Reverse[Table[A[n-k,k],{k,0,n-2}]],{n,2,14}]] (* Stefano Spezia, May 13 2022 *)

A(n,k) = floor(C_n*10^(k+1)) mod 10 where C_n (the base-n Champernowne constant) = Sum_{i>=1} i/(n^(i + Sum_{k=1..i-1} floor(log_n(k+1)))).

A355574 Number of nonnegative integers k with n digits such that x^2 - s*x + p has only integer roots, where s and p denote the sum and product of the digits of k respectively.

Original entry on oeis.org

2, 90, 223, 2686, 31601, 370894, 4220160, 46962379, 512600193

Offset: 1

Stefano Spezia, Jul 07 2022

a(n) is the number of n-digit numbers in A355497.

Cf. A355497.

Cf. A007953, A007954, A063945, A355547.

a(n) <= A063945(n).

Limit_{n->oo} a(n)/a(n-1) = 10.

a(8)-a(9) from Jean-Marc Rebert, Jul 13 2022

cp's OEIS Frontend

A216653 Number A(n,k) of n-digit k-th powers; square array A(n,k), n>=1, k>=1, read by antidiagonals.

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A178500 a(n) = 10^n * signum(n).

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A378564 a(n) is the number of n-digit nonnegative integers with the median of the digits equal to one of the digits.

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A353962 Square array read by descending antidiagonals: The n-th row gives the decimal expansion of the base-n Champernowne constant.

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A355574 Number of nonnegative integers k with n digits such that x^2 - s*x + p has only integer roots, where s and p denote the sum and product of the digits of k respectively.

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