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

1,3

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.

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).

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}.

Example

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.

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 *)

Crossrefs

Cf. A053657.

Sequence in context: A099804 A019596 A088301 * A298903 A127211 A144580

Adjacent sequences: A212426 A212427 A212428 * A212430 A212431 A212432

Keyword

nonn

Author

Jean-Luc Chabert, Jun 21 2012

Status

approved