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%I #114 Jul 02 2023 02:18:35
%S 1,2,24,48,5760,11520,2903040,5806080,1393459200,2786918400,
%T 367873228800,735746457600,24103053950976000,48206107901952000,
%U 578473294823424000,1156946589646848000,9440684171518279680000,18881368343036559360000,271211974879377138647040000
%N a(n) = Product_{p prime} p^{ Sum_{k>=0} floor[(n-1)/((p-1)p^k)]}.
%C LCM of denominators of the coefficients of x^n*z^k in {-log(1-x)/x}^z as k=0..n, as described by triangle A075264.
%C Denominators of integer-valued polynomials on prime numbers (with degree n): 1/a(n) is a generator of the ideal formed by the leading coefficients of integer-valued polynomials on prime numbers with degree less than or equal to n.
%C Also the least common multiple of the orders of all finite subgroups of GL_n(Q) [Minkowski]. Schur's notation for the sequence is M_n = a(n+1). - Martin Lorenz (lorenz(AT)math.temple.edu), May 18 2005
%C This sequence also occurs in algebraic topology where it gives the denominators of the Laurent polynomials forming a regular basis for K*K, the hopf algebroid of stable cooperations for complex K-theory. Several different equivalent formulas for the terms of the sequence occur in the literature. An early reference is K. Johnson, Illinois J. Math. 28(1), 1984, pp.57-63 where it occurs in lines 1-5, page 58. A summary of some of the other formulas is given in the appendix to K. Johnson, Jour. of K-theory 2(1), 2008, 123-145. - Keith Johnson (johnson(AT)mscs.dal.ca), Nov 03 2008
%C a(n) is divisible by n!, by Legendre's formula for the highest power of a prime that divides n!. Also, a(n) is divisible by (n+1)! if and only if n+1 is not prime. - _Jonathan Sondow_, Jul 23 2009
%C Triangle A163940 is related to the divergent series 1^m*1! - 2^m*2! + 3^m*3! - 4^m*4! + ... for m =>-1. The left hand columns of this triangle can be generated with the MC polynomials, see A163972. The Minkowski numbers appear in the denominators of these polynomials. - _Johannes W. Meijer_, Oct 16 2009
%C Unsigned Stirling numbers of the first kind as [s + k, k] (Karamata's notation) where k = {0, 1, 2, ...} and s is in general complex results in Pochhammer[s,k]*(integer coefficient polynomial of (k-1) degree in s) / M[k], where M[k] is the least common multiple of the orders of all finite groups of n x n-matrices over rational numbers (Minkowiski's theorem) which is sequence A053657. - _Lorenz H. Menke, Jr._, Feb 02 2010
%C From _Peter Bala_, Feb 21 2011: (Start)
%C Given a subset S of the integers Z, Bhargava has shown how to associate with S a generalized factorial function, denoted n!_S, which shares many properties of the classical factorial function n!.
%C The present sequence is the generalized factorial function n!_S associated with the set of primes S = {2,3,5,7,...}. The associated generalized exponential function E(x) = Sum_{n>=1} x^(n-1)/a(n) vanishes at x = -2: i.e. Sum_{n>=1} (-2)^n/a(n) = 0.
%C For the table of associated generalized binomial coefficients n!_S/(k!_S*(n-k)!_S) see A186430.
%C This sequence is related to the Bernoulli polynomials in two ways [Chabert and Cahen]:
%C (1) a(n) = (n-1)!*A001898(n-1).
%C (2) (t/(exp(t)-1))^x = sum {n = 0..inf} P(n,x)*t^n/a(n+1),
%C where the P(n,x) are primitive polynomials in the ring Z[x].
%C If p_1,...,p_n are any n primes then the product of their pairwise differences Product_{i<j} (p_i - p_j) is a multiple of a(1)*a(2)*...*a(n-1).
%C (End)
%C LCM of denominators of the coefficients of S(m+n-1,m) as polynomial in m of degree 2*(n-1), as described by triangle A202339. - _Vladimir Shevelev_, Dec 17 2011
%D J.-L. Chabert, S. T. Chapman and S. W. Smith, A basis for the ring of polynomials integer-valued on prime numbers, Factorization in integral domains, Lecture Notes in Pure and Appl. Math. 189, Dekker, New York, 1997.
%D H. Minkowski, Zur Theorie der quadratischen Formen, J. Reine Angew. Math. 101 (1887), 196-202. ( = Ges. Abh., pp. 212-218, Chelsea, New York, 1967.)
%D I. Schur, Über eine Klasse von endlichen Gruppen linearer Substitutionen, Sitzungsber. Preuss. Akad. Wiss. (1905), 77-91. ( = Ges. Abh., Bd. 1, pp. 128-142, Springer-Verlag, Berlin-Heidelberg-New York, 1973.)
%H Gheorghe Coserea, <a href="/A053657/b053657.txt">Table of n, a(n) for n = 1..541</a>
%H F. Bencherif, <a href="http://at.yorku.ca/c/a/y/d/47.htm">Sur une propriété des polynômes de Stirling</a>, 26th Journées Arithmétiques, July 6-10, 2009, Université Jean Monnet, Saint-Etienne, France. [From _Jonathan Sondow_, Jul 23 2009]
%H M. Bhargava, <a href="http://dx.doi.org/10.2307/2695734">The factorial function and generalizations</a>, Amer. Math. Monthly, 107 (2000), 783-799.
%H Paul-Jean Cahen, and J. L. Chabert, <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.123.4.311">What You Should Know About Integer-Valued Polynomials</a>, The American Mathematical Monthly, 123 (No. 4, 2016), 311-337.
%H J.-L. Chabert, <a href="http://dx.doi.org/10.1016/j.ejc.2005.12.009">Integer-valued polynomials on prime numbers and logarithm power expansion</a>, European J. Combinatorics 28 (2007) 754-761. [From _Jonathan Sondow_, Jul 23 2009]
%H J. L. Chabert, <a href="http://www.lamfa.u-picardie.fr/chabert/AboutPolynomials.pdf">About polynomials whose divided differences are integer-valued on prime numbers</a>, ICM 2012 Proceedings, vol. I, pp. 1-7. Complete <a href="http://icm.uaeu.ac.ae/pdf/TomeI.pdf">proceedings</a>. (warning: file size is 26MB). - From _N. J. A. Sloane_, Nov 28 2012
%H J.-L. Chabert and P.-J. Cahen, <a href="https://web.archive.org/web/20120114162802/http://www.latp.cahen.u-3mrs.fr/Recherche/Pubs/oldpb.pdf">Old problems and new questions around integer-valued polynomials and factorial sequences</a>.
%H Robert M. Guralnick and Martin Lorenz, <a href="http://arxiv.org/abs/math/0511191">Orders of Finite Groups of Matrices</a>, arXiv:math/0511191 [math.GR], 2005.
%H K. Johnson, <a href="https://projecteuclid.org/euclid.ijm/1256046153">The action of the stable operations of complex K-theory on coefficient groups</a>, Illinois J. Math. 28(1), 1984, pp. 57-63. [From Keith Johnson (johnson(AT)mscs.dal.ca), Nov 03 2008]
%H K. Johnson, <a href="http://dx.doi.org/10.1017/is008004002jkt042">The invariant subalgebra and anti-invariant submodule of K_*K_{(p)}</a>, Jour. of K-theory 2(1), 2008, 123-145. [From Keith Johnson (johnson(AT)mscs.dal.ca), Nov 03 2008]
%H J.-P. Serre, <a href="https://www.college-de-france.fr/media/jean-pierre-serre/UPL3821667391778701726_6___Bounds_for_the_orders.pdf">Bounds for the orders of the finite subgroups of G(k)</a>, Group Representation Theory (eds. M. Geck, D. Testerman, J. Thevenaz), EPFL Press, Lausanne, 2006, 405-450.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Bhargava_factorial">Bhargava factorial</a>.
%F a(2n) = 2*a(2n-1). - _Jonathan Sondow_, Jul 23 2009
%F a(2*n+1) = 24^n * Product_{i=1..n} A202318(i). - _Vladimir Shevelev_, Dec 17 2011
%F For n>=0, A007814(a(n+1)) = n+A007814(n!). - _Vladimir Shevelev_, Dec 28 2011
%F a(n) = denominator([y^(n-1)] (y/(exp(y)-1))^x). - _Peter Luschny_, May 13 2019
%F Sum_{n>=1} 1/a(n) = A346046. - _Amiram Eldar_, Jul 02 2023
%e a(7)=24^3*Product_{i=1..3} A202318(i)=24^3*1*10*21=2903040. - _Vladimir Shevelev_, Dec 17 2011
%p A053657 := proc(n) local P,p,q,s,r;
%p P := select(isprime,[$2..n]); r:=1;
%p for p in P do s := 0; q := p-1;
%p do if q > (n-1) then break fi;
%p s := s + iquo(n-1,q); q := q*p; od;
%p r := r * p^s; od; r end: # _Peter Luschny_, Jul 26 2009
%p ser := series((y/(exp(y)-1))^x, y, 20): a := n -> denom(coeff(ser, y, n-1)):
%p seq(a(n), n=1..19); # _Peter Luschny_, May 13 2019
%t m = 16; s = Expand[Normal[Series[(-Log[1-x]/x)^z, {x, 0, m}]]];
%t a[n_, k_] := Denominator[ Coefficient[s, x^n*z^k]];
%t Prepend[Apply[LCM, Table[a[n,k], {n,m}, {k,n}], {1}], 1]
%t (* _Jean-François Alcover_, May 31 2011 *)
%t a[n_] := Product[p^Sum[Floor[(n-1)/((p-1) p^k)], {k, 0, n}], {p, Prime[ Range[n] ]}]; Array[a, 30] (* _Jean-François Alcover_, Nov 22 2016 *)
%o (PARI) {a(n)=local(X=x+x^2*O(x^n),D);D=1;for(j=0,n-1,D=lcm(D,denominator( polcoeff(polcoeff((-log(1-X)/x)^z+z*O(z^j),j,z),n-1,x))));return(D)} /* _Paul D. Hanna_, Jun 27 2005 */
%o (PARI) {a(n)=prod(i=1,#factor(n!)~,prime(i)^sum(k=0,#binary(n), floor((n-1)/((prime(i)-1)*prime(i)^k))))} /* _Paul D. Hanna_, Jun 27 2005 */
%o (PARI)
%o S(n, p) = {
%o my(acc = 0, tmp = p-1);
%o while (tmp < n, acc += floor((n-1)/tmp); tmp *= p);
%o return(acc);
%o };
%o a(n) = {
%o my(rv = 1);
%o forprime(p = 2, n, rv *= p^S(n,p));
%o return(rv);
%o };
%o vector(17, i, a(i)) \\ _Gheorghe Coserea_, Aug 24 2015
%Y Cf. A002207, A075264, A075266, A075267.
%Y a(n) = n!*A163176(n). - _Jonathan Sondow_, Jul 23 2009
%Y Cf. A202318.
%Y Appears in A163972. - _Johannes W. Meijer_, Oct 16 2009
%Y Cf. A001898, A007814, A163940, A186430, A202339, A212429, A346046.
%K easy,nonn,nice
%O 1,2
%A _Jean-Luc Chabert_, Feb 16 2000
%E More terms from _Paul D. Hanna_, Jun 27 2005
%E Guralnick and Lorenz link updated by _Johannes W. Meijer_, Oct 09 2009
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