sidef-scripts/Math/largest_factorial_less_than_10_to_the_10_to_the_n.sf

#!/usr/bin/ruby

# Daniel "Trizen" Șuteu
# Date: 14 May 2018
# https://github.com/trizen

# Find the largest number whose factorial is less than 10^(10^n).

# See also:
#   https://oeis.org/A119906

define τ = Num.tau

func a_1(n) {
    var t = n*log(10)
    (n + 10).bsearch_le { |k|
        lngamma(k+1) <=> t
    }
}

func a_2(n) {
    var t = n*log(10)
    (n + 10).bsearch_le { |k|
        ((k*(log(k)-1) + log(τ*k)/2)) <=> t
    }
}

for j in (0..50) {

    var n = 10**j
    var k = a_1(n)

    say "#{j} #{k}"

    assert(lgamma(k+1).log < (j*log(10) + log(log(10))), j)
    assert(lgamma(k+2).log > (j*log(10) + log(log(10))), j)

    assert_eq(k, a_2(n))
}

__END__
0 3
1 13
2 69
3 449
4 3248
5 25205
6 205022
7 1723507
8 14842906
9 130202808
10 1158787577
11 10433891463
12 94851898540
13 869200494599
14 8019346203785
15 74419210652835
16 694100859679691
17 6502464891216879
18 61154108320430275
19 577134533044522749
20 5463531774867094396
21 51865645374019695121
22 493605691759140412264
23 4708412700327664456441
24 45006563676466829747866
25 431030515495800330959690
26 4135271421019159985268144
27 39737872881916155456682466
28 382434585531529136506976638
29 3685654889911271767089158861
30 35566000205751593943612309324
31 343621523511389642125797372225
32 3323659049880171756547149226120
33 32181951201285474537895485226321
34 311917635656773532983489112416836
35 3026031831285250971158863726796844
36 29382559187200956976646535194767034
37 285539923960811790873783241089282065
38 2777061068592571504718455581667927021
39 27028829371643939893024878660179216026
40 263254321831235169282034131531639572407
41 2565752521753016953512727986724591348610
42 25022497618334069614020507552176153613923
43 244179878634545913976035414597821008724201
44 2384185475195871407008689012256184860639887
45 23292152343661431664635019895122205974095819
46 227671082380656404276747365211226185638628432
47 2226510468543990374699303977240445014667122494
48 21784662108910684733496380891610749490017427379
49 213244246864459623913534200247452501697066546234
50 2088314769879577064148725973109379417356880574721