A333402 Numbers m such that the largest digit in the decimal expansion of 1/m is 1.

1, 9, 10, 90, 99, 100, 900, 909, 990, 999, 1000, 9000, 9009, 9090, 9900, 9990, 9999, 10000, 90000, 90009, 90090, 90900, 90909, 99000, 99900, 99990, 99999, 100000, 900000, 900009, 900090, 900900, 909000, 909090, 990000, 990099, 999000, 999900, 999990, 999999, 1000000

Offset: 1

Author

Bernard Schott, Mar 19 2020

Keywords

• nonn
• base

Comments

If m is a term, 10*m is also a term.

If m is a term then m has only digits {1}, {9}, {1,0} or {9,0} in its decimal representation, but this is not sufficient to be a term (see examples).

Some subsequences below (not exhaustive, see crossrefs):

m = 10^k, k >= 0, hence m is in A011557 = {1, 10, 100, 1000, 10000, ...};

m = 9*10^k, k >= 0, hence m is in A052268 = {9, 90, 900, 9000, 90000, ...};

m = 10^k-1, k >= 1, hence m is in A002283 = {9, 99, 999, 9999, 99999, ...};

m = 9*(10^k+1), k >= 1, hence m is in 9*A000533 = {99, 909, 9009, 90009, ...};

m = 9+100*(100^k-1)/11, k >= 0, hence m is in 9*A094028 = {9, 909, 90909, 9090909, ...}.

Examples

			As 1/101 = 0.009900990099..., 101 is not a term.
As 1/909 = 0.001100110011..., 909 is a term.
As 1/9099 = 0.000109902187..., 9099 is not a term.
As 1/9999 = 0.000100010001..., 9999 is also a term.
		

Links

• Index entries for sequences related to decimal expansion of 1/n

Crossrefs

Cf. A333236, A333237 (similar, with 9).

Subsequences: A002283, A011557, A052268.

Subsequences: 9*A000533, 9*A094028, 9*A135577, 9*A261544, 9*A330135.

Programs

Mathematica

Select[Range[10^4], Max @ RealDigits[1/#][[1]] == 1 &] (* Amiram Eldar, Mar 19 2020 *)

Python

from itertools import count, islice
def A333402_gen(startvalue=1): # generator of terms >= startvalue
    for m in count(max(startvalue,1)):
        k = 1
        while k <= m:
            k *= 10
        rset = {0}
        while True:
            k, r = divmod(k, m)
            if max(str(k)) > '1':
                break
            else:
                if r in rset:
                    yield m
                    break
            rset.add(r)
            k = r
            while k <= m:
                k *= 10
A333402_list = list(islice(A333402_gen(),30)) # Chai Wah Wu, Feb 17 2022

Formula

A333236(a(n))= 1.

Extensions

More terms from Jinyuan Wang, Mar 19 2020