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

Offset

2,1

Comments

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

Links

Table of n, a(n) for n=2..12.

Formula

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

Example

a(2) = 3 = 3 whose prime factors in base 2 is: 11.
a(3) = 5 = 5 whose prime factors in base 3 is: 12.
a(4) = 14 = 2*7 whose prime factors in base 4 is: 2, 13.
a(5) = 133 = 7*19 whose prime factors in base 5 is: 12, 34.
a(6) = 706 = 2*353 whose prime factors in base 6 is: 2, 1345.
a(7) = 2490 = 2*3*5*83 whose prime factors in base 7 is: 2, 3, 5, 146.
a(8) = 24258 = 2*3*13*311 whose prime factors in base 8 is: 2, 3, 15, 467.
a(9) = 217230 = 2*3*5*13*557 whose prime factors in base 9 is: 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 is: 2, 3, 5, 18, 67, 49a.
a(12) = 647850030 = 2*3*5*19*1136579 whose prime factors in base 12 is: 2, 3, 5, 17, 4698ab.

Prog

(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)

Crossrefs

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

Sequence in context: A338368 A129326 A167553 * A118562 A273164 A317661

Adjacent sequences: A370609 A370610 A370611 * A370613 A370614 A370615

Keyword

nonn,base,more

Author

Chai Wah Wu, Apr 30 2024

Status

approved