A370612
The smallest number whose prime factor concatenation, when written in base n, does not contain 0 and contains all digits 1,...,(n-1) at least once.
Original entry on oeis.org
3, 5, 14, 133, 706, 2490, 24258, 217230, 2992890, 24674730, 647850030, 4208072190, 82417704810
Offset: 2
a(2) = 3 = 3 whose prime factor in base 2 is: 11.
a(3) = 5 = 5 whose prime factor in base 3 is: 12.
a(4) = 14 = 2*7 whose prime factors in base 4 are: 2, 13.
a(5) = 133 = 7*19 whose prime factors in base 5 are: 12, 34.
a(6) = 706 = 2*353 whose prime factors in base 6 are: 2, 1345.
a(7) = 2490 = 2*3*5*83 whose prime factors in base 7 are: 2, 3, 5, 146.
a(8) = 24258 = 2*3*13*311 whose prime factors in base 8 are: 2, 3, 15, 467.
a(9) = 217230 = 2*3*5*13*557 whose prime factors in base 9 are: 2, 3, 5, 14, 678.
a(10) = 2992890 = 2*3*5*67*1489.
a(11) = 24674730 = 2*3*5*19*73*593 whose prime factors in base 11 are: 2, 3, 5, 18, 67, 49a.
a(12) = 647850030 = 2*3*5*19*1136579 whose prime factors in base 12 are: 2, 3, 5, 17, 4698ab.
a(13) = 4208072190 = 2*3*5*7*61*89*3691 whose prime factors in base 13 are: 2, 3, 5, 7, 49, 6b, 18ac.
a(14) = 82417704810 = 2*3*5*7*23*937*18211 whose prime factors in base 14 are: 2, 3, 5, 7, 19, 4ad, 68cb.
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from math import factorial
from itertools import count
from sympy import primefactors
from sympy.ntheory import digits
def A370612(n): return next(k for k in count(max(factorial(n-1),2)) if 0 not in (s:=set.union(*(set(digits(p,n)[1:]) for p in primefactors(k)))) and len(s) == n-1)
A371511
a(n) is the smallest prime such that its representation in base n contains each of the digits 0,1,...,n-2 at least once and does not contain the digit n-1.
Original entry on oeis.org
3, 73, 683, 8521, 123323, 2140069, 43720693, 1012356487, 26411157737, 749149003087, 23459877380431, 798411310382011, 29471615863458281, 1158045600182881261, 48851274656431280857, 2193475267557861578041, 104737172422274885174411, 5257403213296398892278377
Offset: 3
The corresponding base-n representations are:
n a(n) in base n
------------------------
3 10
4 1021
5 10213
6 103241
7 1022354
8 10123645
9 101236457
10 1012356487
11 10223456798
12 10123459a867
13 1012345678a9b
14 1012345678c9ab
15 1022345678a9cdb
16 10123456789acbed
-
from math import gcd
from sympy import nextprime
from sympy.ntheory import digits
def A371511(n):
m, j = n, 0
if n > 3:
for j in range(1,n-1):
if gcd((n*(n-1)>>1)+j,n-1) == 1:
break
if j == 0:
for i in range(2,n-1):
m = n*m+i
elif j == 1:
for i in range(1,n-1):
m = n*m+i
else:
for i in range(2,1+j):
m = n*m+i
for i in range(j,n-1):
m = n*m+i
m -= 1
while True:
s = digits(m:=nextprime(m), n)[1:]
if n-1 not in s and len(set(s))==n-1:
return m
A371512
a(n) is the smallest prime such that its representation in base n contains each of the digits 1,...,n-2 at least once and does not contain the digit 0 nor the digit n-1.
Original entry on oeis.org
13, 37, 163, 1861, 22481, 304949, 5455573, 112345687, 2831681057, 68057976031, 1953952652167, 61390449569437, 2224884906436873, 77181689614101181, 3052505832274232281, 129003238915759600789, 6090208982148446231753, 276667213296398892309917, 13944042713948404997174231
Offset: 3
The corresponding base-n representations are:
n a(n) in base n
------------------------
3 111
4 211
5 1123
6 12341
7 122354
8 1123465
9 11234567
10 112345687
11 1223456987
12 1123458a967
13 112345678ba9
14 11234567a8bc9
15 122345678acb9d
16 1123456789ceabd
-
from math import gcd
from sympy import nextprime
from sympy.ntheory import digits
def A371512(n):
m, j = 1, 0
if n > 3:
for j in range(1,n-1):
if gcd((n*(n-1)>>1)+j,n-1) == 1:
break
if j == 0:
for i in range(2,n-1):
m = n*m+i
elif j == 1:
for i in range(1,n-1):
m = n*m+i
else:
for i in range(2,1+j):
m = n*m+i
for i in range(j,n-1):
m = n*m+i
m -= 1
while True:
s = digits(m:=nextprime(m), n)[1:]
if (not (0 in s or n-1 in s)) and len(set(s))==n-2:
return m
Showing 1-3 of 3 results.
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