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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-François 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.)