upper-bounds.sf

#!/usr/bin/ruby

# a(n) is the smallest prime p such that omega(p^n - 1) is equal to n, where omega = A001221.
# https://oeis.org/A379450

# Known terms:
#   3, 5, 7, 11, 31, 11, 79, 47, 211, 113, 2473, 47, 45841, 389, 1123, 1061

# Jinyuan Wang: I get a(17) > 729607, a(18) = 373, a(19) > 1600000, a(20)-a(22)=2141, 83071, 43541. Computed with isomega function in https://oeis.org/history?seq=A219019

# PARI/GP program:
#   a(n) = forprime(p=2, oo, if(omega(p^n-1) == n, return(p)));

# Lower-bounds:
#   a(17) > 732181
#   a(19) > 1600000

# Conjectured lower-bounds:
#   a(19) > 1616291
#   a(23) > 198193

local Num!VERBOSE = true
local Num!USE_FACTORDB = true
local Num!USE_CONJECTURES = true

func a(n, from=2) {

    Math.smooth_numbers(2,3,5,7,11,13,17,19,23).each {|k|
        var p = (k+1)

        p > from || next

        if (p.is_prime) {

            var t = (p**n - 1)
            say "[#{n}] Checking k = #{p}: #{t}"

            if (is_omega_prime(t, n)) {
                return p
            }
        }
    }
}

var n = 17
var from = 1272631

#var n = 23
#var from = 1

say a(n, from)