modular_bell_numbers_recurrence.sf

#!/usr/bin/ruby

# A recurrence for computing the Bell numbers modulo a given integer.

# Interesting fact:
#   Bell(p) == 2 (mod p) for prime numbers p.

# See also:
#   https://oeis.org/A166226 -- Bell number n modulo n.
#   https://oeis.org/A325630 -- Numbers k such that Bell(k) is divisible by k.
#   https://mathworld.wolfram.com/BellNumber.html

func modular_binomial(n, k, m) is cached {

    return 1 if (n == k)
    return 0 if (n == 0)

    (__FUNC__(n-1, k-1, m) + __FUNC__(n-1, k, m)) % m
}

func modular_bell(n, k=0, m=n) is cached {

    return 1 if (n == 0)
    return 0 if (k == n)

    (__FUNC__(n, k+1, m) + (__FUNC__(k, 0, m)*modular_binomial(n-1, k, m))) % m
}

for k in (1..20) {
    say "Bell(#{'%2d' % k}) == #{'%2d' % modular_bell(k)} (mod #{k})"
}

__END__
Bell( 1) ==  0 (mod 1)
Bell( 2) ==  0 (mod 2)
Bell( 3) ==  2 (mod 3)
Bell( 4) ==  3 (mod 4)
Bell( 5) ==  2 (mod 5)
Bell( 6) ==  5 (mod 6)
Bell( 7) ==  2 (mod 7)
Bell( 8) ==  4 (mod 8)
Bell( 9) ==  6 (mod 9)
Bell(10) ==  5 (mod 10)
Bell(11) ==  2 (mod 11)
Bell(12) ==  1 (mod 12)
Bell(13) ==  2 (mod 13)
Bell(14) == 12 (mod 14)
Bell(15) ==  5 (mod 15)
Bell(16) ==  3 (mod 16)
Bell(17) ==  2 (mod 17)
Bell(18) == 13 (mod 18)
Bell(19) ==  2 (mod 19)
Bell(20) == 12 (mod 20)