A258981 Numbers containing only 1's and 0's in their base-2, base-3, and base-4 representations.

0, 1, 4, 81, 84, 85, 256, 273, 324, 325, 336, 337, 1089, 1092, 1093, 20496, 20497, 20736, 20737, 20740, 65620, 65856, 65857, 81921, 81984, 81985, 82000, 86032, 86277, 86292, 86293, 86356, 262468, 262480, 263169

Offset

1,3

Comments

As a trend in the first 1000 numbers in the sequence, there tend to be clusters of these numbers, with very large gaps where a number with this property cannot be found.

This sequence lists the numbers that are counted in A230360. - Matthew Goers, Jul 11 2015

Note that a(27) = 82000 also contains no digit > 1 in base 5, see A146025. - Matthew Goers, Jul 11 2015

Numbers that can be expressed both as a sum of distinct powers of 3 and as a sum of distinct powers of 4. - Antti Karttunen, Aug 18 2015

Links

Paul Tek, Table of n, a(n) for n = 1..10000

Paul Tek, PARI program for this sequence

Example

81 is 10000 in base 3 and 1101 in base 4 so 81 is a term.
273 is 101010 in base 3 and 10101 in base 4 so 273 is a term.

Maple

N:= 20: # to get all terms < 2*4^(N-1)
g:= proc(n)
local L, j, m, a;
L:= convert(n, base, 2);
a:= add(4^(j-1)*L[j], j=1..nops(L));
if has(convert(a, base, 3), 2) then NULL else a fi
end proc:
map(g, [$0..2^N]); # Robert Israel, Jul 14 2015

Mathematica

ok3[n_] := 1 == Max@ IntegerDigits[n, 3]; to4[n_] := FromDigits[ IntegerDigits[n, 2], 4]; Select[to4/@ Range[2^20], ok3] (* Giovanni Resta, Jun 16 2015 *)

Prog

(C#)
//The syntax for the EssentialBasic base conversion is as follows:
//string LongDecimalToBase(long initial_number, int base_to_convert_into)
//string BaseToBase(long initial_number, int base_that_initial_number_is_in, int base_to_convert_into
long i = 0; string base3 = ""; string base4 = ""; bool bad = false;
while (true){
   base4 = EssentialBasic.Mathematics.BaseConversion.LongDecimalToBase(i++, 2);
   base3 = EssentialBasic.Mathematics.BaseConversion.BaseToBase(base4, 4, 3);
   for (int i_ = 0; i_ < base3.Length; i_++){
      if (base3.Substring(i_, 1) == "2"){
         bad = true;
         break;
      }
   }
   if (bad == false){
      Console.WriteLine(Convert.ToInt64(EssentialBasic.Mathematics.BaseConversion.BaseToBase(base4, 4, 2), 2));
   }
   bad = false;
}
(Sage) [0]+[n for n in [1..1000000] if max(n.digits(base=3))==1 and max(n.digits(base=4))==1] # Tom Edgar, Jul 11 2015
(PARI) digitsb(m)=vecsort(concat(digits(m, 3), digits(m, 4)), , 8)
is_ok(n)={my(v=digitsb(n), r=0, i); for(i=2, 9, r = r || vecsearch(v, i)); !r}
first(m)={ my(v=vector(m), i, k=0); for(i=1, m, while(!is_ok(k), k++); v[i] = k; k++); v; } /* Anders Hellström, Jul 19 2015 */
(PARI) isok(n) = (n==0) || ((vecmax(digits(n, 3)) < 2) && (vecmax(digits(n, 4)) < 2)); \\ Michel Marcus, Aug 05 2015
(PARI) print1(0); for(n=1, 1e5, vecmax(digits(t=subst(Pol(binary(n)), 'x, 4), 3))<2&&print1(", "t)) \\ M. F. Hasler, Feb 01 2016
(PARI) \\ See links too.
(Python)
def digits(n, b=10): # digits of n in base 2 <= b <= 62
    x, y = n, ''
    while x >= b:
        x, r = divmod(x, b)
        y += str(r) if r < 10 else (chr(r+87) if r < 36 else chr(r+29))
    y += str(x) if x < 10 else (chr(x+87) if x < 36 else chr(x+29))
    return y[::-1]
A258981_list = [n for n in (int(format(d, 'b'), 4) for d in range(10**4)) if max(digits(n, 3)) <= '1'] # Chai Wah Wu, Aug 13 2015

Crossrefs

Intersection of A000695 and A005836.

Cf. A146025, A230360.

Sequence in context: A116189 A159063 A337053 * A072363 A272884 A053901

Adjacent sequences: A258978 A258979 A258980 * A258982 A258983 A258984

Keyword

nonn,base

Author

Phil Lane, Jun 15 2015

Status

approved