experimental-projects/oeis-research/Wesley Ivan Hurt/Sum of denominators sequences/connection_to_cyclotomic_polynomials.sf

#!/usr/bin/ruby

# Sequence by Wesley Ivan Hurt:
#   a(n) = Sum_{i=1..n} denominator(n^i/i).

# OEIS:
#   https://oeis.org/A279911

# Interesting formula:
#   a(p^k) = (p^(2*k+1) + p + 2) / (2*(p+1)), for prime powers p^k.

func a(n) {
    sum(1..n, {|k|
        gcd(n,k) == 1 ? k : k.divisors.last_by {|d| gcd(d,n) == 1 }
    })
}

say 20.range.grep{.is_prime_power}.map(a)
# (p^(2*k+1) + p + 2) / (2*(p+1)),

say 20.range.grep{.is_prime_power}.map {|pp|

    var p = pp.prime_root
    var k = pp.prime_power

    #(p**(2*k + 1) + p + 2) / (2*p + 2)
    k==2 ? (cyclotomic(10, p)+1 / 2) : ((p**(2*k + 1) + p + 2) / (2*p + 2))
}

var k = 14

var a = 20.of {|n|
    (n**(k*2 + 1) + n + 2) / (2*(n+1))
}

for n in (1..100) {
    var t = 20.of {|x|
        cyclotomic(n, x) + 1 / 2
    }

    if (t == a) {
        say "Found: #{n}"
    }
}

__END__

# For k such that 2k+1 is prime,
# the values of n in cyclotomic(n,x) are the even semiprimes.

k =  2 -> (cyclotomic(10, n)+1)/2
k =  3 -> (cyclotomic(14, n)+1)/2
k =  5 -> (cyclotomic(22, n)+1)/2
k =  6 -> (cyclotomic(26, n)+1)/2
k =  8 -> (cyclotomic(34, n)+1)/2
k =  9 -> (cyclotomic(38, n)+1)/2
k = 11 -> (cyclotomic(46, n)+1)/2
k = 14 -> (cyclotomic(58, n)+1)/2


say 20.of{|n|
    #n.is_prime_power ? (cyclotomic(10, n.prime_root)+1 / 2) : a(n)
    #cyclotomic(10, n.prime)+1 / 2


    #n.factor_map {|p|
    #    cyclotomic(10, p)+1 / 2
    #}.sum
}