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

2,1

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

For n <= 36, all terms have prime factors whose concatenation length in base n is n, the minimum possible value. - Dominic McCarty, Jan 07 2025

Links

Table of n, a(n) for n=2..17.

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

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

Cf. A372249, A371993, A027746, A371958, A058909, A185122, A058760.

Sequence in context: A259437 A202737 A186648 * A013033 A376078 A233130

Adjacent sequences: A372306 A372307 A372308 * A372310 A372311 A372312

Keyword

nonn,base,more

Author

Scott R. Shannon, Apr 26 2024

Extensions

a(13)-a(16) from Martin Ehrenstein, May 03 2024

a(17) from Dominic McCarty, Jan 07 2025

Status

approved