A243535 Numbers whose list of divisors contains 2 distinct digits (in base 10).

2, 3, 5, 7, 13, 17, 19, 22, 31, 33, 41, 55, 61, 71, 77, 101, 113, 121, 131, 151, 181, 191, 199, 211, 311, 313, 331, 661, 811, 881, 911, 919, 991, 1111, 1117, 1151, 1171, 1181, 1511, 1777, 1811, 1999, 2111, 2221, 3313, 3331, 4111, 4441, 6661, 7177, 7717, 8111

Offset: 1

Author

Jaroslav Krizek, Jun 13 2014

Keywords

• nonn
• base

Comments

Numbers k such that A037278(k), A176558(k) and A243360(k) contain 2 distinct digits.

Many of the composite terms are in A203897. - Charles R Greathouse IV, Sep 06 2016

Terms are either repdigit numbers (A010785) or contain only 1 and a single other digit. - Michael S. Branicky, Nov 16 2022

Examples

			121 is in the sequence because the list of divisors of 121, i.e., (1, 11, 121), contains 2 distinct digits (1, 2).
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..10000 (first 4317 terms from Robert Israel)

Crossrefs

Cf. A095048, A037278, A176558, A243360.

Sequences of numbers n such that the list of divisors of n contains k distinct digits for 1 <= k <= 10: k = 1: A243534; k = 2: A243535; k = 3: A243536; k = 4: A243537; k = 5: A243538; k = 6: A243539; k = 7: A243540; k = 8: A243541; k = 9: A243542; k = 10: A095050.

Cf. A243543 (the smallest number m whose list of divisors contains n distinct digits).

Programs

Excel

[Row n = 1..10000; Column A: A(n) = A095048(n); Column B: B(n) = IF(A(n)=2;A(n)); Arrangement of column B]

Maple

dmax:= 6: # get all terms of <= dmax digits
Res:= {}:
for a in [0,$2..9] do
    S:= {0}:
    for d from 1 to dmax do
        S:= map(t -> (10*t+1,10*t+a), S);
        Res:= Res union select(filter, S)
    od
od:
sort(convert(Res,list)): # Robert Israel, Sep 05 2016

Mathematica

Select[Range[9000],Length[Union[Flatten[IntegerDigits/@Divisors[ #]]]] == 2&] (* Harvey P. Dale, Dec 14 2017 *)

PARI

isok(n) = vd = []; fordiv(n, d, vd = concat(vd, digits(d))); #Set(vd) == 2; \\ Michel Marcus, Jun 13 2014

Python

from sympy import divisors
from itertools import count, islice, product
def ok(n):
    s = set("1"+str(n))
    if len(s) > 2: return False
    for d in divisors(n, generator=True):
        s |= set(str(d))
        if len(s) > 2: return False
    return len(s) == 2
def agen():
    yield from [2, 3, 5, 7]
    for d in count(2):
        s = set()
        for first, other in product("123456789", "0123456789"):
            for p in product(sorted(set(first+other)), repeat=d-1):
                if other not in p: continue
                t = int(first+"".join(p))
                if ok(t): s.add(t)
        yield from sorted(s)
print(list(islice(agen(), 52))) # Michael S. Branicky, Nov 16 2022