A219365 a(n) = P(n)/(L(n)*P(n/2)*P(n/3)*P(n/7)*P(n/43)*...) with P(n) = floor(n)!, L(n) the LCM of the first n integers and where the sequence 2, 3, 7, 43, ... is A000058.

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 3, 3, 3, 1, 1, 2, 2, 4, 4, 1, 5, 10, 10, 5, 5, 1, 1, 1, 3, 6, 42, 7, 7, 14, 42, 84, 84, 2, 2, 4, 12, 24, 24, 3, 3, 6, 18, 36, 36, 4, 220, 55, 165, 330, 330, 33, 33, 66, 22, 22, 1430, 130, 130, 260, 780, 156, 156, 13

Offset

1,10

Comments

G. Myerson actually proved that P(n)/(P(n/2)*P(n/3)*P(n/7)*P(n/43)*...) is divisible by L(n) in a more general case. That is when n in the above expression is replaced by the terms of a sequence u(n) that satisfies GCD(u(n),u(m))=u(GCD(m,n)). And also when the sequence of quotients q(n)=2,3,7,43,... is replaced by a sequence q(n) such that sum(1/q(n))<=1.

The behavior of a(n) is quite erratic for small values of n, for instance a(26)=10, a(32)=1, a(65)=1430, a(84)=2, a(95)=542640, a(114)=3 (cf. Myerson 1994).

Links

Table of n, a(n) for n=1..72.

G. Bachman, On divisibility properties of certain multinomial coefficients, Journal of Number Theory, Volume 63, Issue 2, April 1997, Pages 244-255.

G. Bachman and T. Kessler, On divisibility properties of certain multinomial coefficients—II, Journal of Number Theory, Volume 106, Issue 1, May 2004, Pages 1-12.

G. Myerson, What the Least Common Multiple Divides, Journal of Number Theory, Volume 48, Issue 1, July 1994, Pages 80-87.

G. Myerson and J. W. Sander, What the Least Common Multiple Divides, II, Journal of Number Theory, Volume 61, Issue 1, November 1996, Pages 67-84.

Prog

(PARI) a(n)=my(t=n!/lcm(vector(n, i, i))/(n\2)!, a1=2, a2=3); while(a2<n, t /= (n\a2)!; [a1, a2]=[a2, a2^2-a2+1]); t \\ Charles R Greathouse IV, Nov 19 2012

Crossrefs

Cf. A000058.

Sequence in context: A201881 A291120 A025485 * A140751 A259922 A162741

Adjacent sequences: A219362 A219363 A219364 * A219366 A219367 A219368

Keyword

nonn

Author

Michel Marcus, Nov 19 2012

Status

approved