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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. 3
1, 1, 2, 4, 48, 96, 1152, 2304, 276480, 552960, 6635520, 13271040, 33443020800, 66886041600, 802632499200, 1605264998400, 385263599616000, 770527199232000, 194172854206464000, 388345708412928000, 512616335105064960000, 1025232670210129920000 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois Alcover, Oct 14 2019, after Alois P. Heinz *)

CROSSREFS

Cf. A053657.

Sequence in context: A099804 A019596 A088301 * A298903 A144580 A144578

Adjacent sequences:  A212426 A212427 A212428 * A212430 A212431 A212432

KEYWORD

nonn

AUTHOR

Jean-Luc Chabert, Jun 21 2012

STATUS

approved

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Last modified September 23 04:54 EDT 2020. Contains 337295 sequences. (Running on oeis4.)