1,2

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)

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.

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.)

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]

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]

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.

a(2n) = 2*a(2n-1). [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Jul 23 2009]

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)

(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)

Cf. A002207, A053657, A075264, A075266, A075267.

a(n) = n!*A163176(n). [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Jul 23 2009]

Appears in A163972. [Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 16 2009]

Sequence in context: A111035 A002552 A075265 this_sequence A079608 A068878 A100918

Adjacent sequences: A053654 A053655 A053656 this_sequence A053658 A053659 A053660

easy,nonn

Jean-Luc Chabert (jlchaber(AT)worldnet.fr), Feb 16 2000

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