OFFSET

0,3

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.

REFERENCES

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

FORMULA

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

EXAMPLE

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.

PROG

(Python 3)

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

(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

CROSSREFS

KEYWORD

nonn,base,fini,full,easy

AUTHOR

Glen Gilchrist, May 20 2021

STATUS

approved