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A135291 Product of the nonzero exponents in the prime factorization of n!. 7
1, 1, 1, 1, 3, 3, 8, 8, 14, 28, 64, 64, 100, 100, 220, 396, 540, 540, 768, 768, 1152, 1944, 4104, 4104, 5280, 7920, 16560, 21528, 31200, 31200, 40768, 40768, 48608, 78120, 161280, 230400, 277440, 277440, 571200, 907200, 1108080, 1108080, 1440504 (list; graph; refs; listen; history; text; internal format)
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

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

Cf. A005361.

Sequence in context: A093366 A204136 A168283 * A058617 A205977 A238623

Adjacent sequences:  A135288 A135289 A135290 * A135292 A135293 A135294

KEYWORD

nonn

AUTHOR

Leroy Quet, Dec 03 2007

EXTENSIONS

More terms from Stefan Steinerberger and R. J. Mathar, Dec 05 2007

STATUS

approved

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Last modified August 23 22:45 EDT 2019. Contains 326254 sequences. (Running on oeis4.)