OFFSET
2,4
COMMENTS
The factorization of n! is n! = 2^T(n,1)*3^T(n,2)*...*p_(pi(n))^T(n,pi(n)) where p_k = k-th prime, pi(n) = A000720(n).
Nonzero terms of A085604; T(n,k) = A085604(n,k), k = 1..A000720(n). - Reinhard Zumkeller, Nov 01 2013
For n=2, 3, 4 and 5, all terms of the n-th row are odd. Are there other such rows? - Michel Marcus, Nov 11 2018
From Gus Wiseman, May 15 2019: (Start)
Differences between successive rows are A067255, so row n is the sum of the first n row-vectors of A067255 (padded with zeros on the right so that all n row-vectors have length A000720(n)). For example, the first 10 rows of A067255 are
{}
1
0 1
2 0
0 0 1
1 1 0
0 0 0 1
3 0 0 0
0 2 0 0
1 0 1 0
with column sums (8,4,2,1), which is row 10.
(End)
For all prime p > 7, 3*p > 2*nextprime(p), so for any n > 21 there will always be a prime p dividing n! with exponent 2 and there are no further rows with all entries odd. - Charlie Neder, Jun 03 2019
LINKS
T. D. Noe, Rows n = 2..300, flattened
H. T. Davis, Tables of the Mathematical Functions, Vols. 1 and 2, 2nd ed., 1963, Vol. 3 (with V. J. Fisher), 1962; Principia Press of Trinity Univ., San Antonio, TX. [Annotated scan of pages 204-208 of Volume 2.] See Table 2 on page 206.
Wenguang Zhai, On the prime power factorization of n!, Journal of Number Theory, Volume 129, Issue 8, August 2009, pages 1820-1836.
FORMULA
T(n,k) = Sum_{i=1..inf} floor(n/(p_k)^i). (Although stated as an infinite sum, only finitely many terms are nonzero.)
T(n,k) = Sum_{i=1..floor(log(n)/log(p_k))} floor(u_i) where u_0 = n and u_(i+1) = floor((u_i)/p_k). - David A. Corneth, Jun 22 2014
EXAMPLE
From Gus Wiseman, May 09 2019: (Start)
Triangle begins:
1
1 1
3 1
3 1 1
4 2 1
4 2 1 1
7 2 1 1
7 4 1 1
8 4 2 1
8 4 2 1 1
10 5 2 1 1
10 5 2 1 1 1
11 5 2 2 1 1
11 6 3 2 1 1
15 6 3 2 1 1
15 6 3 2 1 1 1
16 8 3 2 1 1 1
16 8 3 2 1 1 1 1
18 8 4 2 1 1 1 1
(End)
m such that 5^m||101!: floor(log(101)/log(5)) = 2 terms. floor(101/5) = 20. floor(20/5) = 4. So m = u_1 + u_2 = 20 + 4 = 24. - David A. Corneth, Jun 22 2014
MAPLE
A115627 := proc(n, k) local d, p; p := ithprime(k) ; n-add(d, d=convert(n, base, p)) ; %/(p-1) ; end proc: # R. J. Mathar, Oct 29 2010
MATHEMATICA
Flatten[Table[Transpose[FactorInteger[n!]][[2]], {n, 2, 20}]] (* T. D. Noe, Apr 10 2012 *)
T[n_, k_] := Module[{p, jm}, p = Prime[k]; jm = Floor[Log[p, n]]; Sum[Floor[n/p^j], {j, 1, jm}]]; Table[Table[T[n, k], {k, 1, PrimePi[n]}], {n, 2, 20}] // Flatten (* Jean-François Alcover, Feb 23 2015 *)
PROG
(Haskell)
a115627 n k = a115627_tabf !! (n-2) !! (k-1)
a115627_row = map a100995 . a141809_row . a000142
a115627_tabf = map a115627_row [2..]
-- Reinhard Zumkeller, Nov 01 2013
(PARI) a(n)=my(i=2); while(n-primepi(i)>1, n-=primepi(i); i++); p=prime(n-1); sum(j=1, log(i)\log(p), i\=p) \\ David A. Corneth, Jun 21 2014
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Franklin T. Adams-Watters, Jan 26 2006
STATUS
approved