A055242 -id:A055242 - cp's OEIS frontend

A055240 Number of bases in which n is not divisible by any of its digits.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 3, 0, 3, 0, 1, 1, 6, 0, 3, 1, 2, 0, 7, 0, 7, 1, 4, 4, 7, 0, 9, 4, 6, 1, 11, 0, 13, 2, 3, 6, 17, 0, 11, 3, 8, 3, 18, 2, 13, 3, 11, 9, 22, 0, 18, 9, 8, 4, 15, 1, 23, 8, 16, 5, 24, 1, 24, 12, 11, 8, 24, 4, 29, 4, 15, 14, 31, 1, 22, 14, 21, 8, 34, 1, 23

Offset: 1

Henry Bottomley, May 04 2000

			a(27)=2 because it is written as 27 in base 10 and 25 in base 11 and 27 is not divisible by 2, 5 or 7.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A055238, A055239, A055241, A055242.

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:
map(f, [$1..100]); # Robert Israel, Jan 09 2024

A055239 Numbers which are not divisible by any of their digits in at least one base.

Original entry on oeis.org

11, 13, 17, 19, 21, 22, 23, 25, 26, 27, 29, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92

Offset: 1

Henry Bottomley, May 04 2000

It seems likely all integers greater than 120 appear in this sequence

			9 is excluded because it can be written as 111111111, 1001, 100, 21, 14, 13, 12, 11, 10 or 9 and in every case there is a digit which divides 9; 11 is in the sequence because in base 4 it is written 23 and 11 is not divisible by either 2 or 3

Cf. A038772, A055238-A055242.

A055238 Numbers that are divisible by at least one of their digits in every base.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 24, 28, 30, 36, 42, 48, 60, 120

Offset: 1

Henry Bottomley, May 04 2000

This list is probably finite and complete. For example 180 fails at base 23 and 240 fails at bases 21, 31 and 33

			9 is included because it can be written as 111111111,1001,100,21,14,13,12,11,10 or 9 and in every case there is a digit which divides 9; 11 is not on the list because in base 4 it is written 23 and 11 is not divisible by either 2 or 3.

Cf. A038770, A055239, A055240, A055241, A055242.

divQ[n_, base_] := AnyTrue[Select[IntegerDigits[n, base], Positive], Divisible[n, #]&];
okQ[n_] := AllTrue[Range[2, n-1], divQ[n, #]&];
Select[Range[1000], okQ] (* Jean-François Alcover, Nov 07 2019 *)

A055241 Smallest base in which n is not divisible by any of its digits (0 if no such base).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 5, 0, 0, 0, 5, 0, 5, 0, 8, 6, 5, 0, 7, 7, 10, 0, 6, 0, 7, 9, 7, 6, 4, 0, 7, 7, 7, 11, 7, 0, 4, 12, 16, 7, 4, 0, 9, 11, 9, 9, 5, 10, 8, 10, 10, 9, 4, 0, 8, 8, 11, 11, 5, 14, 5, 9, 9, 11, 9, 13, 5, 10, 11, 10, 5, 10, 5, 11, 11, 11, 10, 15, 5, 10, 10, 13, 5, 19

Offset: 1

Henry Bottomley, May 04 2000

			a(11)=4 because it is written as 111111111111 in base 1, 1011 in base 2, 102 in base 3 and 23 in base 4; 11 is divisible by 1 but not by 2 or 3

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A055238-A055242.

f:= proc(n) local b,L;
  for b from 3 to n-2 do
    L:= convert(convert(n,base,b),set) minus {0};
    if andmap(d -> n mod d <> 0, L) then return b fi
  od;
  0
end proc:
map(f, [$1..100]); # Robert Israel, Oct 29 2024

from sympy.ntheory import digits
def a(n): return next((b for b in range(3, n-2) if not any(n%d==0 for d in digits(n, b)[1:] if d > 0)), 0)
print([a(n) for n in range(1, 91)]) # Michael S. Branicky, Oct 29 2024

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A055240 Number of bases in which n is not divisible by any of its digits.

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A055239 Numbers which are not divisible by any of their digits in at least one base.

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A055238 Numbers that are divisible by at least one of their digits in every base.

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Mathematica

A055241 Smallest base in which n is not divisible by any of its digits (0 if no such base).

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Maple

Python