experimental-projects/oeis-research/Zhao Hui Du/Smallest integer k such that factorial(k) starts with the first n digits of Pi/prog.sf

#!/usr/bin/ruby

# Smallest integer whose factorial's decimal expansion starts with the first n digits of Pi.
# https://oeis.org/A328955

# Known terms:
#   9, 62, 62, 10044, 50583, 1490717, 5573998, 65630447, 688395641, 5777940569, 77773146302, 1154318938997, 1544607046599

define τ = Num.tau
define e = Num.e
define S = τ.sqrt.ln
define T = τ.root(-2*e)

func inverse_factorial_W(n) {
    var l = (n - S)
    l / lambert_w(l / e) - 1/2
}

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

func list() {

    var n = 0
    var min = Inf

    #for (var k = 1; k <= 1e6; ++k) {
    for k in (1..1e6) {

        var t = inverse_factorial_W(log(Num.pi) + log(10)*k)
        var u = abs(t-round(t))

        if (u < min) {
            min = u
            say [k, f(k), min]
        }
    }
}

list();

#~ say f(35835)             #=> 17254
#~ say f(65612)             #=> 17254
#~ say f(215977)            #=> 50583
#~ say f(361229)            #=> 80759
#~ say f(606107)            #=> 129560
#~ say f(18156119873678)    #=> 1544607046599

# 3.1415379577347882738422690467326012034732246681462146178704777124... × 10^215977
# 3.1416026537017914713059250694747809117798457191096278165784809615... × 10^361229
# 3.1416004264160752960646703239185746757645713102358236165630546745... × 10^606107
# 3.14159265358994983986225317956466410382721019457604785880109...      × 10^18156119873678

__END__
[1, 5, 0.170754964162595106192298366446444386187233223018]
[4, 8, 0.119282088619728337069203182464248972832872336214]
[5, 9, 0.0660890277578190984981939385813986170490358621273]
[6, 10, 0.0630814493687193013326830188988446695721598681906]
[14, 17, 0.0442277472832262276235074979061839952357641312835]
[29, 28, 0.0085051071500119547159455753853245768473040300154]
[64, 50, 0.00805287984270237997827138030856530035151899998]
[85, 62, 0.000576891375070655613865104543764424321804876135922]
[1441, 612, 0.000340180561834879812264244917905836193882318073888]
[6146, 2124, 0.0000744365680182018593409222990316358284558204108093]
[6346, 2184, 0.0000585394753736925251787103120793998823716034723696]
[31464, 8945, 0.0000554612592858082825601533548416811887612197770932]
[35479, 9955, 0.0000464069349792776659391879773465441511824005031089]
[35835, 10044, 0.0000144851921972712430203689860099805751717643886668]
[215977, 50583, 0.00000153135254520948990175081082981918126383700418356]
[361229, 80759, 0.000000327373133488305475554506903594881833756188486131]
[606107, 129560, 0.000000237494686233729132768723104874503918154868482228]