A372309 - OEIS

%I #51 Jan 09 2025 19:12:57

%S 2,6,38,174,2866,11670,135570,1335534,15618090,155077890,5148702870,

%T 31771759110,774841780230,11924858870610,253941409789410,

%U 3867805835651310

%N The smallest number whose prime factor concatenation, when written in base n, contains all digits 0,1,...,(n-1).

%C 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)?

%C 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

%C From _Chai Wah Wu_, Apr 29 2024: (Start)

%C a(14) <= 1138370792790 = 2*3*5*7*11*877*561917 whose prime factors in base 14 are: 2, 3, 5, 7, b, 469, 108acd.

%C 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)

%C a(14) <= 774841780230, a(15) <= 11924858870610, a(16) <= 256023548755170, a(17) <= 4286558044897590. - _Daniel Suteu_, Apr 30 2024

%C 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

%H Dominic McCarty, <a href="https://www.youtube.com/watch?v=9SbKjmu-__w">Numbers Whose Prime Factorizations Have Every Digit (OEIS A372309)</a>, YouTube video (2024).

%H Dominic McCarty, <a href="/A372309/a372309.txt">Java program for A372309</a>

%H Dominic McCarty, <a href="/A372309/a372309_1.txt">Bounds on a(n) for n <= 36</a>

%F a(n) >= n!. - _Michael S. Branicky_, Apr 28 2024

%F a(n) <= A185122(n). - _Michael S. Branicky_, Apr 28 2024

%e The factorizations to a(12) are:

%e a(2) = 2 = 10_2, which contains all digits 0..1.

%e a(3) = 6 = 2 * 3 = 2_3 * 10_3, which contain all digits 0..2.

%e a(4) = 38 = 2 * 19 = 2_4 * 103_4, which contain all digits 0..3.

%e a(5) = 174 = 2 * 3 * 29 = 2_5 * 3_5 * 104_5, which contain all digits 0..4.

%e a(6) = 2866 = 2 * 1433 = 2_6 * 10345_6, which contain all digits 0..5.

%e a(7) = 11670 = 2 * 3 * 5 * 389 = 2_7 * 3_7 * 5_7 * 1064_7, which contain all digits 0..6.

%e a(8) = 135570 = 2 * 3 * 5 * 4519 = 2_8 * 3_8 * 5_8 * 10647_8, which contain all digits 0..7.

%e a(9) = 1335534 = 2 * 3 * 41 * 61 * 89 = 2_9 * 3_9 * 45_9 * 67_9 * 108_9, which contain all digits 0..8.

%e a(10) = 15618090 = 2 * 3 * 5 * 487 * 1069, which contain all digits 0..9. See A058909.

%e 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.

%e a(12) = 5148702870 = 2 * 3 * 5 * 151 * 1136579 = 2_12 * 3_12 * 5_12 * 107_12 * 4698ab_12, which contain all digits 0..b.

%o (Python)

%o from math import factorial

%o from itertools import count

%o from sympy import factorint

%o from sympy.ntheory import digits

%o def a(n):

%o     for k in count(factorial(n)):

%o         s = set()

%o         for p in factorint(k): s.update(digits(p, n)[1:])

%o         if len(s) == n: return k

%o print([a(n) for n in range(2, 10)]) # _Michael S. Branicky_, Apr 28 2024

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

%K nonn,base,more

%O 2,1

%A _Scott R. Shannon_, Apr 26 2024

%E a(13)-a(16) from _Martin Ehrenstein_, May 03 2024

%E a(17) from _Dominic McCarty_, Jan 07 2025