A376078
The smallest number whose prime factor concatenation when written in all bases k, where k = 2...n, contains all digits 0,1,...,(k-1).
Original entry on oeis.org
2, 6, 38, 190, 5662, 39255, 1206042, 22481939, 392228153, 6329975006
Offset: 2
a(7) = 39255 as 39255 = 3*5*2617 = 11_2*101_2*101000111001_2 = "11101101000111001" which contains all digits 0...1 = 10_3*12_3*10120221_3 = "101210120221" which contains all digits 0...2, 3_4*11_4*220321_4 = "311220321" which contains all digits 0...3 = 3_5*10_5*40432_5 = "31040432" which contains all digits 0...4 = 3_6*5_6*20041_6 = "3520041" which contains all digits 0...5 = 3_7*5_7*10426_7 = "3510426" which contains all digits 0...6.
A375958
The smallest positive number that when written in all bases 2 to n contains two or more adjacent equal digits.
Original entry on oeis.org
3, 4, 22, 31, 43, 122, 506, 506, 1187, 3199, 11663, 18773, 18815, 32399, 177216, 177216, 177227, 5225284, 5225284, 14295600, 21566225, 36572552, 107730081, 107832239, 952416000, 2293096006, 5587865542, 6587515515, 30174888567, 30174888567, 107824082399
Offset: 2
a(7) = 122 as 122 = 1111010_2 = 11112_3 = 1322_4 = 442_5 = 322_6 = 233_7, each of which contains two or more adjacent equal digits.
A375997
The smallest positive number that when written in all bases 2 to n contains three or more adjacent equal digits.
Original entry on oeis.org
7, 39, 234, 687, 3882, 13999, 196017, 786499, 2790000, 3898880, 150002496, 150002496, 22094555327, 164777407488
Offset: 2
a(7) = 13999 as 13999 = 11011010101111_2 = 201012111_3 = 3122233_4 = 421444_5 = 144451_6 = 55546_7, each of which contains three or more adjacent equal digits.
A375998
The smallest positive number that when written in all bases 2 to n contains four or more adjacent equal digits.
Original entry on oeis.org
15, 80, 4010, 12031, 255312, 2686719, 117178358, 436210878, 91532745555
Offset: 2
a(7) = 2686719 as 2686719 = 1010001111111011111111_2 = 12001111111010_3 = 22033323333_4 = 1141433334_5 = 133330303_6 = 31560000_7, each of which contains four or more adjacent equal digits.
A277028
Numbers that are never pandigital for any base b > 1.
Original entry on oeis.org
0, 1, 3, 7, 31, 255, 32767
Offset: 1
32767 is not pandigital in any base b between 2 and 6:
b 32767 in base b Missing digits
- --------------- --------------
2 111111111111111 0
3 1122221121 0
4 13333333 0, 2
5 2022032 1, 4
6 411411 0, 2, 3, 5
Moreover, 32767 is too small to be pandigital in any base b > 6, hence 32767 is in the sequence.
A351426
a(n) is the smallest number that is (zeroless) pandigital in all bases 2 <= k <= n.
Original entry on oeis.org
1, 5, 45, 198, 4820, 37923, 1021300, 6546092, 514236897, 3166978245, 543912789629, 26110433895907, 1064987485213631, 39225481587293096
Offset: 2
For n = 2, 1 is the smallest base-2 pandigital number.
For n = 3, 5 is the smallest base-3 pandigital number (12) that is also base-2 pandigital (101).
For n = 4, 45 is the smallest base-4 pandigital number (231) that is also base-3 pandigital (1200) and base-2 pandigital (101101).
-
isok(i, n) = {for (b = 2, n, if (Set(digits(i, b))[1] && #Set(digits(i, b)) != b - 1, return (0)); if (Set(digits(i, b))[1] == 0 && #Set(digits(i, b)) != b, return(0)); ); return (1)}
a(n) = {i = n^(n-2); while (! isok(i, n), i++); i; }
-
from itertools import count, product
from sympy.utilities.iterables import multiset_permutations
from gmpy2 import digits, mpz
def A351426(n): # assumes n <= 62
if n == 2:
return 1
dlist = tuple(digits(d,n) for d in range(n))
for l in count(n-2):
for d in range(1,n):
c = None
for t in product(dlist,repeat=l-n+2):
for u in multiset_permutations(sorted(t+dlist[1:d]+dlist[d+1:])):
m = mpz(''.join((dlist[d],)+tuple(u)),n)
for b in range(n-1,1,-1):
if len(set(digits(m,b))|{'0'}) < b:
break
else:
if c != None:
c = min(m,c)
else:
c = m
if c != None:
return int(c) # Chai Wah Wu, Mar 14 2022
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