OFFSET

0,2

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.

LINKS

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

EXAMPLE

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.

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 *)

PROG

(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

CROSSREFS

KEYWORD

nonn,base

AUTHOR

Robert Israel, Jan 09 2024

STATUS

approved