Digit

The number of digits  in an integer  is the number of numbers in some base (usually 10) required to represent it. The numbers 1 to 9 are therefore single digits, while the numbers 10 to 99 are double digits. Terms such as "double-digit inflation" are occasionally encountered, although this particular usage has thankfully not been needed in the U.S. for some time. The number of base- digits in a number  can be calculated as

(1)

where  is the floor function. For , the formula becomes

(2)

The number of digits  in the number  represented in base  is given by the Wolfram Language function DigitCount[n, b, d], with DigitCount[n, b] giving a list of the numbers of each digit in . The total number of digits in a number is given by IntegerLength[n, b].

The positive integers with distinct base-10 digits are given by 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, ... (OEIS A010784). The number of -digit integers is given by

			(3)
			(4)
			(5)
			(6)

where  is a Pochhammer symbol. For , 2, ..., the first few values are 9, 81, 648, 4536, 27216, 136080, 544320, 1632960, 3265920, and 3265920 (OEIS A073531). There are therefore exactly

			(7)
			(8)

such numbers (Pondiczery 1975-a pseudonym for Ralph P. Boas; Foregger 1976), the largest of which is 9876543210.

The sums of the reciprocals of these 8877690 integers (Pondiczery 1975, Foregger 1976) is a rational number with numerator having 14816583 digits and denominator having 14816582 digits and given by

			(9)
			(10)

(OEIS A117914), computed by E. W. Weisstein on Apr. 1, 2006 using gridMathematica.

Numbers in base-10 which are divisible by their digits are 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 15, 22, 24, 33, 36, 44, 48, 55, 66, 77, 88, 99, 111, 112, 115, 122, ... (OEIS A034838). Numbers which are divisible by the sum of their digits are called Harshad numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 18, 20, 21, 24, ... (OEIS A005349). Numbers which are divisible by both their digits and the sum of their digits are 1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 24, 36, 48, 111, 112, 126, 132, 135, 144, ... (OEIS A050104). Numbers which are equal to (i.e., not just divisible by) the product of their divisors and the sum of their divisors are called sum-product numbers and are given by 1, 135, 144, ... (OEIS A038369).

	order	OEIS	Numbers ()
2	increasing
2	nondecreasing	A000225	3, 7, 15, 31, 63, 127, 255, 511, 1023, ...
2	nonincreasing	A023758	2, 3, 4, 6, 7, 8, 12, 14, 15, 16, 24, 28, 30, 31, ...
2	decreasing		2
10	increasing	A009993	12, 13, 14, 15, 16, 17, 18, 19, 23, 24, 25, 26, ...
10	nondecreasing	A009994	11, 12, 13, 14, 15, 16, 17, 18, 19, 22, 23, 24, ...
10	nonincreasing	A009996	10, 11, 20, 21, 22, 30, 31, 32, 33, 40, 41, 42, ...
10	decreasing	A009995	10, 20, 21, 30, 31, 32, 40, 41, 42, 43, 50, 51, ...
16	increasing	A023784	18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, ...
16	nondecreasing	A023757	17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, ...
16	nonincreasing	A023771	17, 32, 33, 34, 48, 49, 50, 51, 64, 65, 66, 67, ...
16	decreasing	A023797	32, 33, 48, 49, 50, 64, 65, 66, 67, 80, 81, 82, ...

In hexadecimal, numbers with increasing digits are called metadromes, those with nondecreasing digits are called plaindrones, those with nonincreasing digits are called nialpdromes, and those with decreasing digits are called katadromes.

The count of numbers with strictly increasing digits in base- is , and the number with strictly decreasing digits is .

REFERENCES:

Bailey, D. H. and Crandall, R. E. "On the Random Character of Fundamental Constant Expansions." Exper. Math. 10, 175-190, 2001. https://www.nersc.gov/~dhbailey/dhbpapers/baicran.pdf.

Foregger, T. "Helping Professor Umbugio. Solution to Problem E2533." Amer. Math. Monthly 83, 570-571, 1976.

Pondiczery, E. S. "Problem E2533." Amer. Math. Monthly 82, 401, 1975.

Sloane, N. J. A. Sequences A005349/M0481, A009993, A009994, A009995, A009996, A010784, A023757, A023758, A023771, A023784, A023797, A034838, A038369, A050104, A073531, and A117914 in "The On-Line Encyclopedia of Integer Sequences."