A372309 The smallest number whose prime factor concatenation, when written in base n, contains all digits 0,1,...,(n-1).
2, 6, 38, 174, 2866, 11670, 135570, 1335534, 15618090, 155077890, 5148702870, 31771759110, 774841780230, 11924858870610, 253941409789410, 3867805835651310
Offset: 2
Examples
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.
Links
- Dominic McCarty, Numbers Whose Prime Factorizations Have Every Digit (OEIS A372309), YouTube video (2024).
- Dominic McCarty, Java program for A372309
- Dominic McCarty, Bounds on a(n) for n <= 36
Programs
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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
Formula
a(n) >= n!. - Michael S. Branicky, Apr 28 2024
a(n) <= A185122(n). - Michael S. Branicky, Apr 28 2024
Extensions
a(13)-a(16) from Martin Ehrenstein, May 03 2024
a(17) from Dominic McCarty, Jan 07 2025
Comments