A335051 a(n) is the smallest decimal number > 1 such that when it is written in all bases from base 2 to base n those numbers all contain both 0 and 1.
2, 9, 19, 28, 145, 384, 1128, 2601, 2601, 101256, 103824, 382010, 572101, 971400, 1773017, 1773017, 22873201, 64041048, 64041048, 1193875201, 2496140640, 10729882801, 21660922801, 120068616277, 333679563001, 427313653201, 427313653201, 10436523921264, 10868368953601
Offset: 2
Examples
a(3) = 9 as 9_2 = 1001 and 9_3 = 100, both of which contain a 0 and 1. a(6) = 145 as 145_2 = 10010001, 145_3 = 12101, 145_4 = 2101, 145_5 = 1040, 145_6 = 401, all of which contain a 0 and 1. a(9) = 2601 as 2601_2 = 101000101001, 2601_3 = 10120100, 2601_4 = 220221, 2601_5 = 40401, 2602_6 = 20013, 2601_7 = 10404, 2601_8 = 5051, 2601_9 = 3510, all of which contain a 0 and 1. Note that, as 2601 also contains a 0 and 1, a(10) = 2601. a(16) = 1773017 as 1773017_2 = 110110000110111011001, 1773017_3 = 10100002010022, 1773017_4 = 12300313121, 1773017_5 = 423214032, 1773017_6 = 102000225, 1773017_7 = 21033101, 1773017_8 = 6606731, 1773017_9 = 3302108, 1773017_10 = 1773017, 1773017_11 = 1001104, 1773017_12 = 716075, 1773017_13 = 4A102C, 1773017_14 = 342201, 1773017_15 = 250512, 1773017_16 = 1B0DD9, all of which contain a 0 and 1.
Links
- Giovanni Resta, Table of n, a(n) for n = 2..32
Programs
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Mathematica
a[n_] := Block[{k=2}, While[ AnyTrue[ Range[n, 2, -1], ! SubsetQ[ IntegerDigits[k, #], {0, 1}] &], k++]; k]; a /@ Range[2, 13] (* Giovanni Resta, May 24 2020 *) -
Python
from numba import njit @njit def hasdigits01(n, b): has0, has1 = False, False while n >= b: n, r = divmod(n, b) if r == 0: has0 = True if r == 1: has1 = True if has0 and has1: return True return has0 and (has1 or n==1) @njit def a(n, start=2): k = start while True: for b in range(n, 1, -1): if not hasdigits01(k, b): break else: return k k += 1 anm1 = 2 for n in range(2, 21): an = a(n, start=anm1) print(an, end=", ") anm1 = an # Michael S. Branicky, Feb 09 2021
Extensions
a(29)-a(30) from Giovanni Resta, May 24 2020
Comments