A153114 -id:A153114 - cp's OEIS frontend

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

Zach J. Shannon and Scott R. Shannon, May 21 2020

The sequence is infinite since 1 + lcm(2,...,n)^2 is always a candidate for a(n). - Giovanni Resta, May 24 2020

			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.

Giovanni Resta, Table of n, a(n) for n = 2..32

Cf. A335066, A258107, A145100, A317725, A181929, A331565, A153114.

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 *)

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

a(29)-a(30) from Giovanni Resta, May 24 2020

A335066 Decimal numbers such that when they are written in all bases from 2 to 10 those numbers all share a common digit (the digit 0 or 1).

Original entry on oeis.org

1, 81, 91, 109, 127, 360, 361, 417, 504, 540, 541, 631, 661, 720, 781, 841, 918, 981, 991, 1008, 1009, 1039, 1080, 1081, 1088, 1089, 1090, 1091, 1093, 1099, 1105, 1111, 1116, 1117, 1118, 1119, 1120, 1121, 1122, 1123, 1124, 1125, 1126, 1128, 1134, 1135, 1136, 1137, 1138, 1139

Offset: 1

Scott R. Shannon and Zach J. Shannon, May 22 2020

As base 2 is included the only possible common digit between all the bases is either a 0 or 1.

			1 is a term as 1 written in all bases is 1.
81 is a term as 81_2 = 1010001, 81_3 = 10000, 81_4 = 1101, 81_5 = 311, 81_6 = 213, 81_7 = 144, 81_8 121, 81_9 = 100, 81_10 = 81, all of which contain the digit 1.
360 is a term as 360_2 = 101101000, 360_3 = 111100, 360_4 = 11220, 360_5 = 2420, 360_6 = 1400, 360_7 = 1023, 360_8 = 550, 360_9 = 550, 360_10 = 360, all of which contain the digit 0.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A335051, A258107, A145100, A317725, A181929, A331565, A153114.

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, True)
    return (has0, has1 or n==1)
def ok(n):
    all0, all1 = True, True
    for b in range(10, 1, -1):
        has0, has1 = hasdigits01(n, b)
        all0 &= has0; all1 &= has1
        if not all0 and not all1: return False
    return all0 or all1
print([k for k in range(1140) if ok(k)]) # Michael S. Branicky, May 23 2022

cp's OEIS Frontend

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.

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