A358854 -id:A358854 - cp's OEIS frontend

A279766 Number of odd digits in the decimal expansions of integers 1 to n.

0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 6, 8, 9, 11, 12, 14, 15, 17, 18, 20, 20, 21, 21, 22, 22, 23, 23, 24, 24, 25, 26, 28, 29, 31, 32, 34, 35, 37, 38, 40, 40, 41, 41, 42, 42, 43, 43, 44, 44, 45, 46, 48, 49, 51, 52, 54, 55, 57, 58, 60, 60, 61, 61, 62, 62, 63, 63, 64, 64

Offset: 0

Joseph Myers, Dec 18 2016

From Bernard Schott, Feb 19 2023: (Start)
Problem 1 of the British Mathematical Olympiad, round 1, in 2016/2017 asked: when the integers 1, 2, 3, ..., 2016 are written down in base 10, how many of the digits in the list are odd? The answer is a(2016) = 4015.
The similar sequence but with number of even digits is A358854. (End)

Joseph Myers, Table of n, a(n) for n = 0..1000
United Kingdom Mathematics Trust, 2016/17 British Mathematical Olympiad Round 1, Problem 1.
Index to sequences related to Olympiads.

Cf. A007908, A058183, A117804, A196564, A358854.

a:= proc(n) option remember; `if`(n=0, 0, a(n-1)+
      nops(select(x-> x::odd, convert(n,base,10))))
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Dec 22 2016

Table[Count[Flatten@ IntegerDigits@ Range[0, n], d_ /; OddQ@ d], {n, 0, 68}] (* or *)
Accumulate@ Table[Count[IntegerDigits@ n, d_ /; OddQ@ d], {n, 0, 68}] (* Michael De Vlieger, Dec 22 2016 *)

a(n) = A196564(A007908(n)). - Michel Marcus, Dec 18 2016

a(n) = A117804(n+1) - A358854(n) (number of total digits - number of even digits). - Bernard Schott, Feb 19 2023

A358439 Number of even digits necessary to write all positive n-digit integers.

Original entry on oeis.org

4, 85, 1300, 17500, 220000, 2650000, 31000000, 355000000, 4000000000, 44500000000, 490000000000, 5350000000000, 58000000000000, 625000000000000, 6700000000000000, 71500000000000000, 760000000000000000, 8050000000000000000, 85000000000000000000, 895000000000000000000

Offset: 1

Bernard Schott, Nov 16 2022

If nonnegative n-digit integers were considered, then a(1) would be 5.
Also, a(n) is the total number of holes in all positive n-digit integers, assuming 4 has no hole. Digits 0, 6 and 9 have 1 hole, digit 8 has 2 holes, and other digits have no holes or (circular) loops (as in A064532).
Proof of the first formula: For n>=2, to write all positive n-digit integers, digits 6, 8, 9 occur A081045(n-1) = (9n+1)*10^(n-2) times each, and digit 0 occurs A212704(n-1) = 9*(n-1)*10^(n-2) times; so a(n) = 4*A081045(n-1) + A212704(n-1).
For a(1), if 0 were included then there would be 5 holes in the 1-digit numbers 0..9.

			To write the integers from 10 up to 99, each of the digits 2, 4, 6 and 8 must be used 19 times, and digit 0 must be used 9 times hence a(2) = 4*19 + 9 = 85.

Index entries for linear recurrences with constant coefficients, signature (20,-100).

Cf. A064532, A081045, A212704.

Cf. A113119, A196563, A358854, A359271.

Maple
```
seq((5*(9*n-1))*10^(n-2), n = 1 .. 30);
```

a[n_] := 5*(9*n - 1)*10^(n - 2); Array[a, 22] (* Amiram Eldar, Nov 16 2022 *)

a(n) = 5*(9n-1)*10^(n-2).
Formulas coming from the name with even digits:
a(n) = A358854(10^n-1) - A358854(10^(n-1)-1).
a(n) = A113119(n) - A359271(n) for n >= 2.
Formula coming from the comment with holes:
a(n) = Sum_{k=10^(n-1)..10^n-1} A064532(k).

cp's OEIS Frontend

A279766 Number of odd digits in the decimal expansions of integers 1 to n.

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