OFFSET
2,1
COMMENTS
Up to a(12) all terms have prime factors whose concatenation length in base n is n, the minimum possible value. Is this true for all a(n)?
a(13) <= 31771759110 = 2*3*5*7*13*61*190787 whose prime factors in base 13 are: 2, 3, 5, 7, 10, 49, 68abc. Sequence is a subsequence of A058760. - Chai Wah Wu, Apr 28 2024
From Chai Wah Wu, Apr 29 2024: (Start)
a(14) <= 1138370792790 = 2*3*5*7*11*877*561917 whose prime factors in base 14 are: 2, 3, 5, 7, b, 469, 108acd.
a(15) <= 23608327052310 = 2*3*5*7*11*13*233*3374069 whose prime factors in base 15 are: 2, 3, 5, 7, b, d, 108, 469ace. (End)
a(14) <= 774841780230, a(15) <= 11924858870610, a(16) <= 256023548755170, a(17) <= 4286558044897590. - Daniel Suteu, Apr 30 2024
FORMULA
a(n) >= n!. - Michael S. Branicky, Apr 28 2024
a(n) <= A185122(n). - Michael S. Branicky, Apr 28 2024
EXAMPLE
The factorizations to a(12) are:
a(2) = 2 = 10_2, which contains all digits 0..1.
a(3) = 6 = 2 * 3 = 2_3 * 10_3, which contain all digits 0..2.
a(4) = 38 = 2 * 19 = 2_4 * 103_4, which contain all digits 0..3.
a(5) = 174 = 2 * 3 * 29 = 2_5 * 3_5 * 104_5, which contain all digits 0..4.
a(6) = 2866 = 2 * 1433 = 2_6 * 10345_6, which contain all digits 0..5.
a(7) = 11670 = 2 * 3 * 5 * 389 = 2_7 * 3_7 * 5_7 * 1064_7, which contain all digits 0..6.
a(8) = 135570 = 2 * 3 * 5 * 4519 = 2_8 * 3_8 * 5_8 * 10647_8, which contain all digits 0..7.
a(9) = 1335534 = 2 * 3 * 41 * 61 * 89 = 2_9 * 3_9 * 45_9 * 67_9 * 108_9, which contain all digits 0..8.
a(10) = 15618090 = 2 * 3 * 5 * 487 * 1069, which contain all digits 0..9. See A058909.
a(11) = 155077890 = 2 * 3 * 5 * 11 * 571 * 823 = 2_11 * 3_11 * 5_11 * 10_11 * 47a_11 * 689_11, which contain all digits 0..a.
a(12) = 5148702870 = 2 * 3 * 5 * 151 * 1136579 = 2_12 * 3_12 * 5_12 * 107_12 * 4698ab_12, which contain all digits 0..b.
PROG
(Python)
from math import factorial
from itertools import count
from sympy import factorint
from sympy.ntheory import digits
def a(n):
for k in count(factorial(n)):
s = set()
for p in factorint(k): s.update(digits(p, n)[1:])
if len(s) == n: return k
print([a(n) for n in range(2, 10)]) # Michael S. Branicky, Apr 28 2024
CROSSREFS
KEYWORD
nonn,base,more
AUTHOR
Scott R. Shannon, Apr 26 2024
EXTENSIONS
a(13)-a(16) from Martin Ehrenstein, May 03 2024
STATUS
approved