A368930 a(n) is the least k which is divisible by none of its digits in exactly n bases, or 0 if there is no such k.

1, 11, 27, 17, 33, 70, 23, 29, 51, 37, 92, 41, 74, 43, 82, 65, 69, 47, 53, 136, 106, 87, 59, 67, 71, 154, 172, 118, 111, 79, 146, 83, 123, 378, 89, 97, 153, 212, 125, 101, 103, 121, 119, 107, 113, 225, 250, 127, 159, 206, 202, 218, 155, 183, 143, 131, 137, 139, 161, 1020, 151, 169, 157, 149, 286

Offset: 0

Author

Robert Israel, Jan 09 2024

Keywords

• nonn
• base

Comments

a(n) is the least k, if it exists, such that A055240(k) = n.

It appears that a(n) = 0 for n = 159, 208, 266, 267, 328, 405, 484, 492, ....

Entries of 0 in the a-file are conjectural: if they are not 0, they are > 35000.

Examples

			a(3) = 17 because there are exactly 3 bases in which 17 is divisible by none of its digits: these bases are 5, 6, 7, because 17 = 32_5 = 25_6 = 23_7, and 17 is not divisible by any of the digits 2, 3 and 5 from these bases.  In every other base, 17 is divisible by at least one of its digits; e.g., in base 10, 17 is divisible by 1.  And 17 is the first number for which there are exactly 3 such bases.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..1000. Entries of 0 are conjectural.

Crossrefs

Cf. A055240.

Programs

Maple

f:= proc(n)
   nops(select(b -> not ormap(d -> d <> 0 and n mod d = 0, convert(n,base,b)), [$3 .. (n-1)/2]))
end proc:
V:= Array(0..100): count:= 0:
for n from 1 while count < 101 do
v:= f(n);
if v <= 100 and V[v] = 0 then V[v]:= n; count:= count+1 fi;
od:
convert(V,list);

Mathematica

isDiv[k_, b_] := Module[{d}, d = IntegerDigits[k, b]; Or @@ (Mod[k, #] == 0 & /@ DeleteCases[d, 0])];
co[k_] := co[k] = Module[{c = 0, b = 2}, While[b <= k, If[Not[isDiv[k, b]], c++]; b++]; c];
a[n_] := a[n] = Module[{k = 1}, While[co[k] != n, k++;]; k];
Table[a[n], {n, 0, 64}] (* Robert P. P. McKone, Jan 10 2024 *)

Python

from itertools import count, islice
from sympy.ntheory.factor_ import digits
def agen():
    adict, n = dict(), 0
    for k in count(1):
        v = sum(1 for i in range(2, k) if all(d==0 or k%d for d in digits(k, i)[1:]))
        if v not in adict: adict[v] = k
        while n in adict: yield adict[n]; n += 1
print(list(islice(agen(), 65))) # Michael S. Branicky, Jan 10 2024