A275600 Numbers that can be written in all bases from base 2 to base 6 using only the digits 0, 1 and 2.

0, 1, 2, 6, 36, 37, 260, 1302, 1376, 1380, 1381, 1382, 1556, 1560, 1561, 1562, 16932, 562500, 562501, 562502, 562506, 562512, 562536, 562537, 562752, 562760, 23610752, 23610756, 23610757, 23610786, 23615750, 23615760, 23615761, 23615762, 23615785, 23615786, 23626310

Offset

1,3

Comments

Is there any number that keeps this property also in base 7, other than the trivial cases 0,1,2?

Links

Rémy Sigrist and Chai Wah Wu, Table of n, a(n) for n = 1..10000 [Terms 1 through 187 by Chai Wah Wu]

Example

16932 is in the sequence because this number can be written in bases 2 through 6 using only the digits 0, 1 and 2: 16932(b4)  = 10020210 / (b5) = 1020212 / (b6) = 210220.

Mathematica

Select[Range[10^6], Function[k, Max@ Flatten@ Map[IntegerDigits[k, #] &, Range[4, 6]] < 3]] (* or *)
Select[Range[10^5], Function[k, Total@ Flatten@ Map[Take[RotateRight@ DigitCount[k, #], -(# - 3)] &, Range[4, 6]] == 0]] (* (not as efficient) Michael De Vlieger, Aug 03 2016 *)

Prog

(Python) from gmpy2 import digits
A275600_list = [n for n in (int(digits(m, 3), 6) for m in range(10**6)) if max(digits(n, 5)) <= '2' and max(digits(n, 4)) <= '2'] # Chai Wah Wu, Aug 15 2016
(Perl) use ntheory ":all"; my($x, $n10)=(0, 0); while ($x < 50) { my $n = fromdigits( todigitstring($n10++, 3), 6);  next if vecany { $_ > 2 } todigits($n, 4);  next if vecany { $_ > 2 } todigits($n, 5);  print ++$x, " $n\n"; } # Dana Jacobsen, Aug 16 2016
(PARI) nextWithSmallDigits(n, base) = my (pow=1, rem=n, val=0, d); while (rem>0, d = rem % base; rem = rem \ base; if (d>2, val = 0; rem = rem+1, val = val + d*pow); pow = pow * base); return (val)
{ n = 0; prev = 0; while (n < 300, succ = prev; for (b=4, 6, succ = nextWithSmallDigits(succ, b)); if (prev==succ, n = n+1; print(n " " prev); prev = succ+1, prev = succ)) } \\ Rémy Sigrist, Sep 08 2016

Crossrefs

Cf. A146025, A146026, A146027, A146028, A146029, A258107, A258981.

Sequence in context: A247209 A324577 A324582 * A226368 A058083 A034526

Adjacent sequences: A275597 A275598 A275599 * A275601 A275602 A275603

Keyword

nonn,base

Author

Sergio Pimentel, Aug 03 2016

Extensions

a(18)-a(26) from Michael De Vlieger, Aug 03 2016

a(27)-a(37) from Chai Wah Wu, Aug 15 2016

Status

approved