A060238 a(n) = det(M) where M is an n X n matrix with M[i,j] = lcm(i,j).

1, 1, -2, 12, -48, 960, 11520, -483840, 3870720, -69672960, -2786918400, 306561024000, 7357464576000, -1147764473856000, -96412215803904000, -11569465896468480000, 185111454343495680000, -50350315581430824960000, -1812611360931509698560000

Offset: 0

Author

MCKAY john (mckay(AT)cs.concordia.ca), Mar 21 2001

Keywords

• sign

References

• J.-M. De Koninck & A. Mercier, 1001 Problèmes en Théorie Classique des Nombres, Problème 695, pp. 90, 297-298, Ellipses, Paris, 2004.
• J. Sandor and B. Crstici, Handbook of Number Theory II, Springer, 2004, p. 265, eq. 10.

Links

• Enrique Pérez Herrero, Table of n, a(n) for n = 0..200
• Index to sequences related to determinants.

Crossrefs

Cf. A000142, A001088, A013939, A023900, A048803, A060239, A085542.

Programs

Maple

A060238:=n->n!*mul((1-ithprime(i))^floor(n/ithprime(i)), i=1..numtheory[pi](n)): seq(A060238(n), n=0..20); # Wesley Ivan Hurt, Aug 15 2016

Mathematica

A060238[n_]:=n!*Product[(1 - Prime[i])^Floor[n/Prime[i]], {i, PrimePi[n]}]; Array[A060238, 20] (* Enrique Pérez Herrero, Jun 08 2010 *)

PARI

a(n)=n!*prod(p=1,sqrtint(n),if(isprime(p),(1-p)^floor(n/p),1)) \\ Benoit Cloitre, Jan 31 2008

Formula

For n >= 2, a(n) = n! * Product_{j=2..n} Product_{p|j} (1-p) (where the second product is over all primes p that divide j) (cf. A023900). - Avi Peretz (njk(AT)netvision.net.il), Mar 22 2001

a(n) = n! * Product_{p<=n} (1-p)^floor(n/p) where the product runs through the primes. - Benoit Cloitre, Jan 31 2008

a(n) = A000142(n) * A085542(n). - Enrique Pérez Herrero, Jun 08 2010

a(n) = A001088(n) * A048803(n) * (-1)^A013939(n). - Amiram Eldar, Dec 19 2018

a(n) = Product_{k=1..n} (-1)^A001221(k) * A000010(k) * A007947(k) [De Koninck & Mercier]. - Bernard Schott, Dec 11 2020

Extensions

a(0)=1 prepended by Alois P. Heinz, Jan 25 2023