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.

3, 5, 14, 133, 706, 2490, 24258, 217230, 2992890, 24674730, 647850030, 4208072190, 82417704810

Offset: 2

Author

Chai Wah Wu, Apr 30 2024

Keywords

• nonn
• base
• more

Comments

All terms are squarefree. Many thanks to Michael Branicky for pointing out errors in the terms in the original submission.

Examples

			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.
		

Crossrefs

Cf. A371194, A372309, A372249, A371993, A027746, A371958, A058909, A185122.

Programs

Python

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)

Formula

(n-1)! <= a(n) <= A371194(n).

Extensions

a(13)-(14) from Dominic McCarty, Jan 07 2025