A055240 -id:A055240 - cp's OEIS frontend

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

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 *)

A384211 a(n) is the number of distinct ways of representing n in any integer base >= 2 using only prime digits.

Original entry on oeis.org

0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 2, 2, 2, 1, 4, 2, 3, 2, 3, 1, 6, 2, 3, 4, 4, 1, 6, 2, 5, 4, 5, 2, 7, 2, 7, 4, 5, 3, 8, 4, 9, 3, 7, 3, 12, 3, 6, 5, 6, 4, 11, 2, 9, 4, 9, 6, 13, 3, 11, 8, 12, 3, 12, 3, 13, 7, 8, 5, 14, 5, 13, 5, 11, 4, 15, 3, 13, 8, 10, 7, 15

Offset: 0

Felix Huber, May 23 2025

The representations of n remain the same for bases greater than n, as they all consist solely of the digit n.

			The a(43) = 9 representations of [4,3] in base 10 using only prime digits are [2,2,3] in base 4, [5,3] in base 8, [3,7] in base 12, [2,13] in base 15, [2,11] in base 16, [2,7] in base 18, [2,5] in base 19, [2,3] in base 20 and [43] in bases >= 44.

Felix Huber, Table of n, a(n) for n = 0..10000

Cf. A055240, A355034.

A384211:=proc(n)
    local a,b,c;
    a:=0;
    for b from 2 to n+1 do
        c:=convert(n,'base',b);
        if select(isprime,c)=c then
            a:=a+1
        fi
    od;
    return a
end proc;
seq(A384211(n),n=0..87);
A384211representations:=proc(n)
    local L,b,c;
    L:=[];
    for b from 2 to n+1 do
        c:=convert(n,'base',b);
        if select(isprime,c)=c then
            L:=[op(L),b,ListTools:-Reverse(c)]
        fi
    od;
    return op(L)
end proc;
A384211representations(43);

a[n_] := Boole[PrimeQ[n]] + Count[Range[2, n-1], ?(AllTrue[IntegerDigits[n, #], PrimeQ] &)]; Array[a, 100 ,0] (* _Amiram Eldar, May 23 2025 *)

a(n) = sum(k=2, n+1, my(d=digits(n, k)); #select(isprime, d) == #d); \\ Michel Marcus, May 26 2025

from sympy import isprime
from sympy.ntheory import digits
def a(n): return len(set(t for b in range(2, n+2) if all(map(isprime, (t:=tuple(digits(n, b)[1:]))))))
print([a(n) for n in range(84)]) # Michael S. Branicky, May 23 2025

A384212 a(n) is the number of bases >= 2 in which the alternating sum of digits of n is equal to 0.

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 1, 2, 2, 2, 1, 4, 1, 2, 3, 3, 1, 4, 1, 3, 2, 2, 1, 6, 2, 2, 3, 4, 1, 6, 1, 4, 3, 2, 2, 7, 1, 2, 3, 6, 1, 5, 1, 4, 5, 2, 1, 8, 2, 3, 3, 4, 1, 5, 2, 6, 3, 2, 1, 9, 1, 2, 5, 5, 3, 6, 1, 4, 2, 6, 1, 10, 1, 2, 5, 4, 2, 5, 1, 8, 3, 2, 1, 8, 3, 2, 2

Offset: 1

Felix Huber, May 24 2025

The alternating sum of digits of n is equal to 1 for base n and equal to n for bases > n.

			a(72) = 10 because the alternating sum of digits of n is equal to 0 in the 10 bases 2 [1, 0, 0, 1, 0, 0, 0], 3 [2, 2, 0, 0], 5 [2, 4, 2], 7 [1, 3, 2], 8 [1, 1, 0], 11 [6, 6], 17 [4, 4], 23 [3, 3], 35 [2, 2] and 71 [1, 1].

Felix Huber, Table of n, a(n) for n = 1..10000

Cf. A055240, A061845, A225693, A135499, A135551, A384211.

A384212:=proc(n)
    local a,b,c,i;
    a:=0;
    for b from 2 to n-1 do
        c:=convert(n,'base',b);
    	   if add(c[i]*(-1)^i,i=1..nops(c))=0 then
           a:=a+1
        fi
    od;
    return a
end proc;
seq(A384212(n),n=1..87);
A384212bases:=proc(n)
    local L,b,c,i;
    L:=[];
    for b from 2 to n-1 do
        c:=convert(n,'base',b);
    	if add(c[i]*(-1)^i,i=1..nops(c))=0 then
            L:=[op(L),b,ListTools:-Reverse(c)]
        fi
    od;
    return op(L)
end proc;
A384212bases(72);

q[n_, b_] := Module[{d = IntegerDigits[n, b]}, Sum[(-1)^k*d[[k]], {k, 1, Length[d]}] == 0 ]; a[n_] := Count[Range[2, n-1], ?(q[n, #] &)]; Array[a, 100] (* _Amiram Eldar, May 24 2025 *)

a(n) = sum(b=2, n-1, my(d=digits(n, b)); sum(k=1, #d, (-1)^k*d[k]) == 0); \\ Michel Marcus, May 24 2025

from sympy.ntheory import digits
def s(v): return sum(v[::2]) - sum(v[1::2])
def a(n): return sum(1 for b in range(2, n) if s(digits(n, b)[1:]) == 0)
print([a(n) for n in range(1, 87)]) # Michael S. Branicky, May 24 2025

Trivial bounds: 1 <= a(n) <= n - 2 for n >= 3 because the representation of n in base n-1 is [1,1] and the alternating sum of digits of n is > 0 for bases >= n.

A054476 Numbers not divisible by any of their digits when written in base 5.

Original entry on oeis.org

13, 17, 19, 23, 53, 65, 67, 73, 77, 79, 85, 89, 94, 95, 97, 98, 103, 113, 115, 118, 119, 253, 263, 265, 269, 313, 317, 319, 323, 325, 329, 335, 337, 343, 347, 349, 353, 365, 367, 373, 377, 379, 385, 389, 394, 395, 397, 398, 425, 427, 437, 439, 443, 445, 449

Offset: 1

Henry Bottomley, May 12 2000

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A038772, A055239, A055240.

is(n)=my(d=Set(digits(n,5))); for(i=if(d[1],1,2),#d, if(n%d[i]==0, return(0))); 1 \\ Charles R Greathouse IV, Feb 23 2017

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.

Original entry on oeis.org

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

Robert Israel, Jan 09 2024

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.

			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.

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

Cf. A055240.

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);

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 *)

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

A384436 a(n) is the number of distinct ways to represent n in any integer base >= 2 using only square digits.

Original entry on oeis.org

1, 1, 1, 2, 4, 3, 3, 3, 3, 6, 5, 4, 5, 5, 4, 4, 6, 5, 4, 5, 7, 7, 5, 5, 7, 8, 6, 6, 8, 7, 7, 7, 7, 7, 7, 6, 11, 9, 6, 7, 10, 7, 7, 7, 8, 8, 8, 6, 8, 11, 7, 7, 9, 10, 7, 7, 10, 10, 7, 7, 11, 10, 7, 7, 13, 11, 7, 7, 11, 10, 7, 7, 10, 11, 8, 8, 11, 11, 9, 8, 11, 15

Offset: 0

Felix Huber, May 29 2025

The representations of n remain the same for bases greater than n, as they all consist solely of the digit n.

			The a(36) = 11 distinct ways to represent 36 using only square digits are [1,0,0,1,0,0] in base 2, [1,1,0,0] in base 3, [1,0,0] in base 6, [4,4] in base 8, [4,0] in base 9, [1,16] in base 20, [1,9] in base 27, [1,4] in base 32, [1,1] in base 35, [1,0] in base 36 and [36] in bases >= 37.

Felix Huber, Table of n, a(n) for n = 0..10000

Cf. A046030, A055240, A061845, A077268, A126071, A135551, A384211, A384212.

A384436:=proc(n)
    local a,b,c;
    a:=0;
    for b from 2 to n+1 do
        c:=convert(n,'base',b);
        if select(issqr,c)=c then
            a:=a+1
        fi
    od;
    return max(1,a)
end proc;
seq(A384436(n),n=0..81);

a[n_] := Sum[Boole[AllTrue[IntegerDigits[n, b], IntegerQ[Sqrt[#]] &]], {b, 2, n+1}]; a[0] = 1; Array[a, 100, 0] (* Amiram Eldar, May 29 2025 *)

a(n) = sum(b=2, n+1, my(d=digits(n,b)); #select(issquare, d) == #d); \\ Michel Marcus, May 29 2025

Trivial lower bound for n >= 2: a(n) >= 2 for nonsquares n and a(n) >= 3 for squares n because in base 2 the representations of n consists only of the square digits '0' and '1', in base n the representation of n is [1,0] and in bases > n the representation of n is [n].

A054475 Numbers not divisible by any of their digits when written in base 4.

Original entry on oeis.org

11, 35, 43, 47, 59, 131, 139, 143, 163, 175, 179, 187, 191, 203, 227, 235, 239, 251, 515, 523, 527, 547, 559, 563, 571, 575, 643, 655, 683, 691, 703, 707, 715, 719, 739, 751, 755, 763, 767, 779, 803, 811, 815, 827, 899, 907, 911, 931, 943, 947, 955, 959

Offset: 1

Henry Bottomley, May 12 2000

Cf. A038772, A055239, A055240.

A054478 Numbers not divisible by any of their digits when written in base 6.

Original entry on oeis.org

17, 22, 23, 29, 34, 77, 89, 101, 107, 113, 130, 131, 137, 142, 143, 149, 161, 166, 167, 173, 174, 178, 179, 197, 202, 203, 209, 214, 437, 449, 461, 467, 509, 521, 527, 533, 539, 557, 563, 569, 581, 593, 599, 611, 617, 629, 641, 647, 653, 670, 671, 677, 682

Offset: 1

Henry Bottomley, May 12 2000

Cf. A038772, A055239, A055240.

A054484 Numbers not divisible by any of their digits when written in base 7.

Original entry on oeis.org

17, 19, 23, 25, 26, 31, 33, 34, 37, 38, 39, 41, 46, 47, 101, 103, 115, 117, 119, 121, 125, 131, 133, 137, 139, 143, 149, 151, 152, 161, 163, 167, 173, 175, 178, 179, 181, 182, 187, 188, 191, 193, 194, 199, 201, 202, 217, 221, 223, 227, 229, 230, 231, 233, 237

Offset: 1

Henry Bottomley, May 12 2000

Cf. A038772, A055239, A055240.

A054864 Numbers not divisible by any of their digits when written in base 8.

Original entry on oeis.org

19, 21, 23, 29, 31, 35, 37, 38, 39, 43, 46, 47, 53, 55, 59, 61, 62, 131, 133, 135, 149, 151, 155, 157, 163, 167, 173, 179, 181, 183, 187, 191, 197, 199, 211, 215, 221, 223, 227, 229, 230, 232, 238, 239, 247, 248, 251, 253, 254, 259, 261, 262, 263, 275, 277

Offset: 1

Henry Bottomley, May 12 2000

Cf. A038772, A055239, A055240.

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

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A384211 a(n) is the number of distinct ways of representing n in any integer base >= 2 using only prime digits.

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A384212 a(n) is the number of bases >= 2 in which the alternating sum of digits of n is equal to 0.

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A054476 Numbers not divisible by any of their digits when written in base 5.

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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.

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A384436 a(n) is the number of distinct ways to represent n in any integer base >= 2 using only square digits.

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A054475 Numbers not divisible by any of their digits when written in base 4.

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A054478 Numbers not divisible by any of their digits when written in base 6.

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