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

Author

Michel Marcus, Nov 19 2012

Keywords

• nonn

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

• 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.

Crossrefs

Cf. A000058.

Programs

PARI

a(n)=my(t=n!/lcm(vector(n,i,i))/(n\2)!,a1=2,a2=3);while(a2Charles R Greathouse IV, Nov 19 2012