A053657 - OEIS

OFFSET

1,2

COMMENTS

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.

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. - 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 (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

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} x^(n-1)/a(n) vanishes at x = -2: i.e. Sum_{n>=1} (-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. - Vladimir Shevelev, Dec 17 2011

Sometimes called "Minkowski numbers" (e.g., by Guralnick and Lorenz), after the German mathematician Hermann Minkowski (1864-1909). - Amiram Eldar, Aug 24 2024

REFERENCES

Jean-Luc Chabert, Scott T. Chapman, and William W. Smith, A basis for the ring of polynomials integer-valued on prime numbers, in: Daniel Anderson (ed.), Factorization in integral domains, Lecture Notes in Pure and Appl. Math. 189, Dekker, New York, 1997.

LINKS

Gheorghe Coserea, Table of n, a(n) for n = 1..541

F. Bencherif, Sur une propriété des polynômes de Stirling, 26th Journées Arithmétiques, July 6-10, 2009, Université Jean Monnet, Saint-Etienne, France. [From Jonathan Sondow, Jul 23 2009]

M. Bhargava, The factorial function and generalizations, Amer. Math. Monthly, 107 (2000), 783-799.

Paul-Jean Cahen, and J. L. Chabert, What You Should Know About Integer-Valued Polynomials, The American Mathematical Monthly, 123 (No. 4, 2016), 311-337.

J.-L. Chabert, Integer-valued polynomials on prime numbers and logarithm power expansion, European J. Combinatorics 28 (2007) 754-761. [From Jonathan Sondow, Jul 23 2009]

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). - From N. J. A. Sloane, Nov 28 2012

J.-L. Chabert and P.-J. Cahen, Old problems and new questions around integer-valued polynomials and factorial sequences.

Robert M. Guralnick and Martin Lorenz, Orders of Finite Groups of Matrices, in: William Chin, James Osterburg and Declan Quinn (eds.), Groups, Rings and Algebras, Contemporary Mathematics, Vol. 420 (2006), pp. 141-161; arXiv preprint, arXiv:math/0511191 [math.GR], 2005; author's link.

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]

Hermann Minkowski, Zur Theorie der quadratischen Formen, J. Reine Angew. Math. 101 (1887), 196-202. ( = Ges. Abh., pp. 212-218, Chelsea, New York, 1967.)

Issai 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.)

J.-P. Serre, Bounds for the orders of the finite subgroups of G(k), Group Representation Theory (eds. M. Geck, D. Testerman, J. Thevenaz), EPFL Press, Lausanne, 2006, 405-450.

Wikipedia, Bhargava factorial.

FORMULA

a(2n) = 2*a(2n-1). - 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

a(n) = denominator([y^(n-1)] (y/(exp(y)-1))^x). - Peter Luschny, May 13 2019

Sum_{n>=1} 1/a(n) = A346046. - Amiram Eldar, Jul 02 2023

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 := 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: # Peter Luschny, Jul 26 2009

ser := series((y/(exp(y)-1))^x, y, 20): a := n -> denom(coeff(ser, y, n-1)):

seq(a(n), n=1..19); # Peter Luschny, May 13 2019

MATHEMATICA

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]

(* Jean-François Alcover, May 31 2011 *)

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 *)

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)} /* Paul D. Hanna, Jun 27 2005 */

(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 */

(PARI)

S(n, p) = {

my(acc = 0, tmp = p-1);

while (tmp < n, acc += floor((n-1)/tmp); tmp *= p);

return(acc);

};

a(n) = {

my(rv = 1);

forprime(p = 2, n, rv *= p^S(n, p));

return(rv);

};