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 A053657 a(n) = Product_{p prime} p^{ Sum_{k>= 0} floor[(n-1)/((p-1)p^k)]}. 37
 1, 2, 24, 48, 5760, 11520, 2903040, 5806080, 1393459200, 2786918400, 367873228800, 735746457600, 24103053950976000, 48206107901952000, 578473294823424000, 1156946589646848000, 9440684171518279680000, 18881368343036559360000, 271211974879377138647040000 (list; graph; refs; listen; history; text; internal format)
 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..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=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 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); }; 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 A292162 Adjacent sequences: A053654 A053655 A053656 * A053658 A053659 A053660 KEYWORD easy,nonn,nice AUTHOR Jean-Luc 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

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Last modified December 7 11:43 EST 2022. Contains 358656 sequences. (Running on oeis4.)