OFFSET
0,5
COMMENTS
a(n) = A005361(n!). For n >= 2, a(n) = the number of positive divisors of n! which themselves are each divisible by every prime <= n. For p = any prime, a(p) = a(p-1). a(0)=a(1)=1 because the product of the exponents is over the empty set.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
FORMULA
EXAMPLE
6! = 720 has a prime factorization of 2^4 * 3^2 * 5^1. So a(6) = 4*2*1 = 8.
Also, 720 is divisible by a(6)=8 positive divisors which themselves are each divisible by every prime <= 6 (i.e., are each divisible by 2*3*5 = 30): 30, 60, 90, 120, 180, 240, 360, 720.
MAPLE
A005361 := proc(n) mul( op(2, i), i=ifactors(n)[2]) ; end: A135291 := proc(n) A005361(n!) ; end: seq(A135291(n), n=0..50) ; # R. J. Mathar, Dec 12 2007
MATHEMATICA
Table[Product[FactorInteger[n! ][[i, 2]], {i, 1, Length[FactorInteger[n! ]]}], {n, 0, 50}] (* Stefan Steinerberger, Dec 05 2007 *)
Table[Times@@Transpose[FactorInteger[n!]][[2]], {n, 0, 50}] (* Harvey P. Dale, Aug 16 2011 *)
PROG
(PARI) valp(n, p)=my(s); while(n\=p, s+=n); s
a(n)=my(s=1); forprime(p=2, n\2, s*=valp(n, p)); s \\ Charles R Greathouse IV, Oct 09 2016
CROSSREFS
KEYWORD
nonn
AUTHOR
Leroy Quet, Dec 03 2007
EXTENSIONS
More terms from Stefan Steinerberger and R. J. Mathar, Dec 05 2007
STATUS
approved