

A053657


a(n) = Prod_{p prime} p^{ Sum_{k>= 0} floor[(n1)/((p1)p^k)]}.


29



1, 2, 24, 48, 5760, 11520, 2903040, 5806080, 1393459200, 2786918400, 367873228800, 735746457600, 24103053950976000, 48206107901952000, 578473294823424000, 1156946589646848000, 9440684171518279680000
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OFFSET

1,2


COMMENTS

LCM of denominators of the coefficients of x^n*z^k in {log(1x)/x}^z as k=0..n, as described by triangle A075264.
Denominators of integervalued polynomials on prime numbers (with degree n): 1/a(n) is a generator of the ideal formed by the leading coefficients of integervalued polynomials on prime numbers with degree less than or equal to n.
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
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 Ktheory. 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.5763 where it occurs in lines 15, page 58. A summary of some of the other formulas is given in the appendix to K. Johnson, Jour. of Ktheory 2(1), 2008, 123145.  Keith Johnson (johnson(AT)mscs.dal.ca), Nov 03 2008
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
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
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 (k1) degree in s) / M[k], where M[k] is the least common multiple of the orders of all finite groups of n x nmatrices over rational numbers (Minkowiski's theorem) which is sequence A053657.  Lorenz H. Menke, Jr., Feb 02 2010
From Peter Bala, Feb 21 2011: (Start)
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!.
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..inf} x^(n1)/a(n) vanishes at x = 2: i.e. sum {n = 1..inf} (2)^n/a(n) = 0.
For the table of associated generalized binomial coefficients n!_S/(k!_S*(nk)!_S) see A186430.
This sequence is related to the Bernoulli polynomials in two ways [Chabert and Cahen]:
(1) a(n) = (n1)!*A001898(n1).
(2) (t/(exp(t)1))^x = sum {n = 0..inf} P(n,x)*t^n/a(n+1),
where the P(n,x) are primitive polynomials in the ring Z[x].
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(n1).
(End)
LCM of denominators of the coefficients of S(m+n1,m) as polynomial in m of degree 2*(n1), as described by triangle A202339.  Vladimir Shevelev, Dec 17 2011
From Lorenz H. Menke, Jr., Feb 02 2010: (Start)
(Other) Using LaTex we have a nice format for the sequence representation:
$M \left({n}\right) = \prod\limits_{p \in prime}^{}
{p}^{\left\lfloor{\frac{n}{p  1}}\right\rfloor
+ \left\lfloor{\frac{n}{p \left({p  1}\right)}}\right\rfloor
+ \left\lfloor{\frac{n}{{p}^{2} \left({p  1}\right)}}\right\rfloor
+ \cdots}$ (End)


REFERENCES

PaulJean Cahen, JL Chabert, What You Should Know About IntegerValued Polynomials, The American Mathematical Monthly, 123 (No. 4, 2016), 311337.
J.L. Chabert, S. T. Chapman and S. W. Smith, A basis for the ring of polynomials integervalued on prime numbers, Factorization in integral domains, Lecture Notes in Pure and Appl. Math. 189, Dekker, New York, 1997.
K. Johnson, The action of the stable operations of complex Ktheory on coefficient groups, Illinois J. Math. 28(1), 1984, pp.5763. [From Keith Johnson (johnson(AT)mscs.dal.ca), Nov 03 2008]
H. Minkowski, Zur Theorie der quadratischen Formen, J. Reine Angew. Math. 101 (1887), 196202. ( = Ges. Abh., pp. 212218, Chelsea, New York, 1967.)
I. Schur, Über eine Klasse von endlichen Gruppen linearer Substitutionen, Sitzungsber. Preuss. Akad. Wiss. (1905), 7791. ( = Ges. Abh., Bd. 1, pp. 128142, SpringerVerlag, BerlinHeidelbergNew York, 1973.)


LINKS

Gheorghe Coserea, Table of n, a(n) for n = 1..541
F. Bencherif, Sur une propriete des polynomes de Stirling [From Jonathan Sondow, Jul 23 2009]
M. Bhargava, The factorial function and generalizations, Amer. Math. Monthly, 107(2000), 783799.
J.L. Chabert, Integervalued polynomials on prime numbers and logarithm power expansion, European J. Combinatorics 28 (2007) 754761. [From Jonathan Sondow, Jul 23 2009]
J. L. Chabert, About polynomials whose divided differences are integervalued on prime numbers, ICM 2012 Proceedings, vol. I, pp. 17. Complete proceedings (warning: file size is 26MB).  From N. J. A. Sloane, Nov 28 2012
J.L. Chabert and P.J. Cahen, Old problems and new questions around integervalued polynomials and factorial sequences
Robert M. Guralnick and Martin Lorenz, Orders of Finite Groups of Matrices, arXiv:math/0511191.
K. Johnson, The invariant subalgebra and antiinvariant submodule of K_*K_{(p)}, Jour. of Ktheory 2(1), 2008, 123145. [From Keith Johnson (johnson(AT)mscs.dal.ca), Nov 03 2008]


FORMULA

a(2n) = 2*a(2n1).  Jonathan Sondow, Jul 23 2009
a(2*n+1) = 24^n*Product_{i=1..n} A202318(i).  Vladimir Shevelev, Dec 17 2011
For n>=0, A007814(a(n+1)) = n+A007814(n!).  Vladimir Shevelev, Dec 28 2011


EXAMPLE

a(7)=24^3*Product_{i=1..3} A202318(i)=24^3*1*10*21=2903040.  Vladimir Shevelev, Dec 17 2011


MAPLE

A053657 := proc(n) local P, p, q, s, r;
P := select(isprime, [$2..n]); r:=1;
for p in P do s := 0; q := p1;
do if q > (n1) then break fi;
s := s + iquo(n1, q); q := q*p; od;
r := r * p^s; od; r end: # Peter Luschny, Jul 26 2009


MATHEMATICA

m = 16; s = Expand[Normal[Series[(Log[1x]/x)^z, {x, 0, m}]]];
a[n_, k_] := Denominator[ Coefficient[s, x^n*z^k]];
Prepend[Apply[LCM, Table[a[n, k], {n, m}, {k, n}], {1}], 1]
(* JeanFrançois Alcover, May 31 2011 *)
a[n_] := Product[p^Sum[Floor[(n1)/((p1) p^k)], {k, 0, n}], {p, Prime[ Range[n] ]}]; Array[a, 30] (* JeanFrançois Alcover, Nov 22 2016 *)


PROG

(PARI) {a(n)=local(X=x+x^2*O(x^n), D); D=1; for(j=0, n1, D=lcm(D, denominator( polcoeff(polcoeff((log(1X)/x)^z+z*O(z^j), j, z), n1, x)))); return(D)} /* Paul D. Hanna, Jun 27 2005 */
(PARI) {a(n)=prod(i=1, #factor(n!)~, prime(i)^sum(k=0, #binary(n), floor((n1)/((prime(i)1)*prime(i)^k))))} /* Paul D. Hanna, Jun 27 2005 */
(PARI)
S(n, p) = {
my(acc = 0, tmp = p1);
while (tmp < n, acc += floor((n1)/tmp); tmp *= p);
return(acc);
};
a(n) = {
my(rv = 1);
forprime(p = 2, n, rv *= p^S(n, p));
return(rv);
};
vector(17, i, a(i)) \\ Gheorghe Coserea, Aug 24 2015


CROSSREFS

Cf. A002207, A075264, A075266, A075267.
a(n) = n!*A163176(n).  Jonathan Sondow, Jul 23 2009
Cf. A202318.
Appears in A163972.  Johannes W. Meijer, Oct 16 2009
Cf. A001898, A186430, A212429.
Sequence in context: A249277 A002552 A075265 * A079608 A257663 A068878
Adjacent sequences: A053654 A053655 A053656 * A053658 A053659 A053660


KEYWORD

easy,nonn,nice


AUTHOR

JeanLuc Chabert, Feb 16 2000


EXTENSIONS

More terms from Paul D. Hanna, Jun 27 2005
Guralnick and Lorenz link updated by Johannes W. Meijer, Oct 09 2009


STATUS

approved



