A212429 a(n) is the LCM of denominators of polynomials of degree n which are integer-valued on primes together with their first divided differences.

1, 1, 2, 4, 48, 96, 1152, 2304, 276480, 552960, 6635520, 13271040, 33443020800, 66886041600, 802632499200, 1605264998400, 385263599616000, 770527199232000, 194172854206464000, 388345708412928000, 512616335105064960000, 1025232670210129920000

Offset: 1

Author

Jean-Luc Chabert, Jun 21 2012

Keywords

• nonn

Comments

a(n) is also the n-th Bhargava's factorial n_P^{{1}} of the set P of primes with respect to the first divided difference.

Examples

			a(5) = 48 because f(x) = (x-1)(x-2)(x-3)(x-5)(x-7)/48 satisfies f(p) and (f(p)-f(q))/(p-q) are integers for all primes p,q.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..170
• M. Bhargava, On P-orderings, Integer-Valued Polynomials, and Ultrametric Analysis, J. Amer. Math. Soc., 22 (2009), 963-993.
• J. L. Chabert, About polynomials whose divided differences are integer-valued on prime numbers, ICM 2012 Proceedings, vol. I, pp. 1-7. Complete proceedings (warning: file size is 26MB).

Crossrefs

Cf. A053657.

Programs

Maple

a:= proc(n) local i, p, wp, r;
      r:=1;
      for i do p:= ithprime(i);
        wp:= p^(w(p,n-1));
        if wp=1 then break fi;
        r:= r*wp
      od; r
    end:
w:= proc(p, n) local d, k, r;
      r:= 0;
      for k from 0 do d:= floor(n/((p-1)*p^k));
        if d=0 then break fi;
        r:= r+d;
      od;
      r -t(n,p)
    end:
t:= proc(n, p) local h, q;
      q:= n/(p-1);
      for h from 0 while q>= p^h do od; h
    end:
seq (a(n), n=1..30);  # Alois P. Heinz, Jun 25 2012

Mathematica

a[n_] := Module[{i, p, wp, r}, r = 1; For[i = 1, True, i++, p = Prime[i]; wp = p^w[p, n - 1]; If[wp == 1, Break[]]; r = r*wp]; r];
w[p_, n_] := Module[{d, k, r}, r = 0; For[k = 0, True, k++, d = Floor[n/((p - 1)*p^k)]; If[d == 0, Break[]]; r = r + d]; r - t[n, p]];
t[n_, p_] := Module[{h, q}, q = n/(p - 1); For[h = 0, q >= p^h , h++]; h];
a /@ Range[1, 30] (* Jean-François Alcover, Oct 14 2019, after Alois P. Heinz *)

Formula

a(n) = Prod_{p prime} p^w_p(n-1) where w_p(n) = Sum_{k>=0} floor(n / ((p-1)*p^k)) - t_{p,n} and p^(t_{p,n}-1) <= n/(p-1) < p^t_{p,n}.