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A053657
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a(n) = Prod_{p prime} p^{ Sum_{k>= 0} floor[(n-1)/((p-1)p^k)]}.
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26
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1, 2, 24, 48, 5760, 11520, 2903040, 5806080, 1393459200, 2786918400, 367873228800, 735746457600, 24103053950976000, 48206107901952000, 578473294823424000, 1156946589646848000, 9440684171518279680000
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| LCM of denominators of the coefficients of x^n*z^k in {-ln(1-x)/x}^z as k=0..n, as described by triangle A075264.
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.
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 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. [From 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. [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Jul 23 2009]
Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 16 2009: (Start)
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.
(End)
From Lorenz H. Menke, Jr. (lnz2004(AT)mindspring.com), Feb 02 2010: (Start)
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.
(End)
From Peter Bala, 21 Feb 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^(n-1)/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*(n-k)!_S) see A186430.
This sequence is related to the Bernoulli polynomials in two ways [Chabert and Cahen]:
(1) a(n) = (n-1)!*A001898(n-1).
(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(n-1).
(End)
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. [From Vladimir Shevelev, Dec 17 2011]
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REFERENCES
| J.-L. Chabert, Integer-valued polynomials on prime numbers and logarithm power expansion, European J. Combinatorics 28 (2007) 754-761. [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Jul 23 2009]
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.
K. Johnson, The action of the stable operations of complex K-theory on coefficient groups, Illinois J. Math. 28(1), 1984, pp.57-63. [From Keith Johnson (johnson(AT)mscs.dal.ca), Nov 03 2008]
K. Johnson, The invariant subalgebra and anti-invariant submodule of $K_*K_{(p)}$, Jour. of K-theory 2(1), 2008, 123-145. [From Keith Johnson (johnson(AT)mscs.dal.ca), Nov 03 2008]
H. Minkowski, Zur Theorie der quadratischen Formen, J. Reine Angew. Math. 101 (1887), 196-202. ( = Ges. Abh., pp. 212-218, Chelsea, New York, 1967.)
I. Schur, Ueber 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.)
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LINKS
| F. Bencherif, Sur une propriete des polynomes de Stirling [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Jul 23 2009]
Robert M. Guralnick and Martin Lorenz, Orders of Finite Groups of Matrices, arXiv:math/0511191.
M. Bhargava, The factorial function and generalizations, Amer. Math. Monthly, 107(2000), 783-799.
J-L. Chabert and P-J. Cahen, Old problems and new questions around integer-valued polynomials and factorial sequences
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FORMULA
| a(2n) = 2*a(2n-1). [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Jul 23 2009]
a(2*n+1)=24^n*prod{i=1,...,n}A202318(i). [From Vladimir Shevelev, Dec 17 2011]
For n>=0, A007814(a(n+1))=n+A007814(n!). [From Vladimir Shevelev, Dec 28 2011]
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EXAMPLE
| a(7)=24^3*prod{i=1,2,3}A202318(i)=24^3*1*10*21=2903040. [From Vladimir Shevelev, Dec 17 2011]
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MAPLE
| Contribution from Peter Luschny (peter(AT)luschny.de), Jul 26 2009: (Start)
A053657 := proc(n) local P, p, q, s, r;
P := select(isprime, [$2..n]); r:=1;
for p in P do s := 0; q := p-1;
do if q > (n-1) then break fi;
s := s + iquo(n-1, q); q := q*p; od;
r := r * p^s; od; r end: (End)
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MATHEMATICA
| (* Contribution from Jean-François Alcover, May 31 2011: (Start) *)
m = 16; s = Expand[Normal[Series[(-Log[1-x]/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] (* End *)
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PROG
| (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)} (Hanna)
(PARI) {a(n)=prod(i=1, #factor(n!)~, prime(i)^sum(k=0, #binary(n), floor((n-1)/((prime(i)-1)*prime(i)^k))))} (Hanna)
Contribution from Lorenz H. Menke, Jr. (lnz2004(AT)mindspring.com), 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)
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CROSSREFS
| Cf. A002207, A053657, A075264, A075266, A075267.
a(n) = n!*A163176(n). [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Jul 23 2009], A202318
Appears in A163972. [Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 16 2009]
A001898, A186430
Sequence in context: A111035 A002552 A075265 * A079608 A068878 A100918
Adjacent sequences: A053654 A053655 A053656 * A053658 A053659 A053660
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KEYWORD
| easy,nonn
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AUTHOR
| Jean-Luc Chabert (jlchaber(AT)worldnet.fr), Feb 16 2000
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EXTENSIONS
| More terms from Paul D. Hanna (pauldhanna(AT)juno.com), Jun 27 2005
Guralnick and Lorenz link updated by Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 09 2009
Replaced arXiv URL by non-cached variant - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Oct 23 2009
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