A344474 Least number k such that half of the numbers from 0 to k inclusive contain the digit n.

1, 1, 2915, 39365, 472391, 590489, 6377291, 7440173, 8503055, 9565937

Offset: 0

Author

Glen Gilchrist, May 20 2021

Keywords

• nonn
• base
• fini
• full
• easy

Comments

"Half-numbers" are those for which half of the numbers including and preceding it contain a specific digit.

For each digit there are a finite number of nonnegative integers k such that exactly half of the numbers from 0 to k contain the digit. This sequence gives the first of these.

Examples

			a(0)=1 since among the numbers 0,1 exactly half contain a digit "0" and 1 is the smallest number where this occurs.
a(1)=1 since among the numbers 0,1 exactly half contain a digit "1" and 1 is the smallest number where this occurs.
a(2)=2915 since among the numbers 0,1,2,...,2915 exactly half contain a digit "2" and 2915 is the smallest number where this occurs.
a(3)=39365 since among the numbers 0,1,2,...,39365 exactly half contain a digit "3" and 39365 is the smallest number where this occurs.
		

References

• Andrew Hilton, 101 Puzzles to Solve on your Microcomputer, 1984, HARRAP, page 57.

Crossrefs

Cf. A016189, A344634 (half-zero sequence), A344636 (half-one sequence).

Programs

PARI

a(n)={if(n>=1&&n<10, my(k=0); while(n*(2*9^k-10^k)>10^k, k++); 2*9^k*n - 1, n==0)} \\ Andrew Howroyd, May 25 2021

Python

for z in range (0, 10):
    z_s = str(z)
    counts=0
    for x in range (0,1000000000):
        x_s = str(x)
        if z_s in x_s:
            counts += 1
        if counts / (x+1) == 0.5:
            print(x)
            break

Formula

a(n) == 1457 (mod 1458) for n >= 2. - Hugo Pfoertner, May 25 2021