A226443 Number of distinct shadow transforms for sequences of length n.

1, 1, 1, 3, 12, 48, 288, 1356, 10848, 70896, 588480

Offset: 0

Author

Charles R Greathouse IV, Jun 06 2013

Keywords

• nonn
• hard
• more

Examples

			The sequence (i, j, k) has shadow transform (0, 1, m) where m is the number of even numbers in {i, j}, so a(3) = 3.
		

Links

• Lorenz Halbeisen and Norbert Hungerbuehler, Number theoretic aspects of a combinatorial function, Notes on Number Theory and Discrete Mathematics 5 (1999), 138-150.
• Lorenz Halbeisen, A number-theoretic conjecture and its implication for set theory, Acta Math. Univ. Comenianae 74(2) (2005), 243-254.
• OEIS Wiki, Shadow transform.

Crossrefs

Cf. A000522, A056793.

Programs

PARI

sh(v)=vector(#v,i,my(n=i-1);sum(j=1,n,v[j]%n==0));
a(n)={
  my(L=log(n+.5), t=primes(primepi(n)), D=divisors(prod(i=1,#t, t[i]^(L\log(t[i])))), nd=#D, v=[]);
    for(i=1,nd^(n-1),
        my(s=sh(vector(n,j,D[i\nd^(j-1)%nd+1])));
        if(!setsearch(v,s),
            v=vecsort(concat(v,[s]))
        )
    );
    #v
};

PARI

v=[]; fordiv(72,a, fordiv(72,b, fordiv(72,c, fordiv(72,d, fordiv(72,e, fordiv(72,f, fordiv(72,g, fordiv(72,h, fordiv(9,i, u=sh([a,b,c,d,e,f,g,h,i,0]); if(!vecsearch(v,u), v=vecsort(concat(v,[u])))))))))))); (5+1)*(7+1)*#v \\ computes a(10)

Formula

a(p+1) = (p+1)a(p) where p is prime.

a(n-1) <= a(n) <= n*a(n-1).