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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, 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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified August 8 19:29 EDT 2020. Contains 336298 sequences. (Running on oeis4.)