A034876 Number of ways to write n! as a product of smaller factorials each greater than 1.

0, 0, 0, 1, 0, 1, 0, 1, 1, 2, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Author

Erich Friedman

Keywords

• easy
• nonn
• nice

Comments

By definition, a(n) > 0 if and only if n is a member of A034878. If n > 2, then a(n!) > max(a(n), a(n!-1)), as (n!)! = n!*(n!-1)!. Similarly, a(A001013(n)) > 0 for n > 2. Clearly a(n)=0 if n is a prime A000040. So a(n+1)=1 if n=2^p-1 is a Mersenne prime A000668, as (n+1)!=(2!)^p*n! and n is prime. - Jonathan Sondow, Dec 15 2004

From Antti Karttunen, Dec 25 2018: (Start)

If n! = a! * x! * y! * ... * z!, with a > x >= y >= z, then A006530(n!) = A006530(a!) > A006530(x!). This follows because all rows in A115627 end with 1, that is, because all factorials >= 2 are in A102750.

If all the two-term solutions are of the form n! = a! * x! = b! * y! = ... = c! * z! (that is, all are products of two factorials larger than one), with a > x, b > y, ..., c > z, then a(n) = (a(x)+1 + a(y)+1 + ... + a(z)+1).

Values 0..5 occur for the first time at n = 1, 4, 10, 576, 13824, 69120.

In range 1..69120 differs from A322583 only at positions n = 1, 2, 9, 10 and 16.

(End)

Examples

			a(10) = 2 because 10! = 3! * 5! * 7! = 6! * 7! are the only two ways to write 10! as a product of smaller factorials > 1.
From _Antti Karttunen_, Dec 25 2018: (Start)
a(8) = 1 because 8! = 7! * (2!)^3.
a(9) = 1 because 9! = 7! * 3! * 3! * 2!.
a(16) = 2 because 16! = 15! * (2!)^4 = 14! * 5! * 2!.
a(144) = 2 because 144! = 143! * 4! * 3! = 143! * 3! * 3! * 2! * 2!.
a(576) = 3 because 576! = 575! * 4! * 4! = 575! * 4! * 3! * 2! * 2! = 575! * 3! * 3! * 2! * 2! * 2! * 2!.
a(720) = 2 because 720! = 719! * 6! = 719! * 5! * 3!.
a(3456) = 3 because 3456! = 3455! * 4! * 4! * 3! = 3455! * 4! * 3! * 3! * 2! * 2! = 3455! * 3! * 3! * 3! * 2! * 2! * 2! * 2!.
(End)
		

References

• R. K. Guy, Unsolved Problems in Number Theory, B23.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..69120
• Eric Weisstein's World of Mathematics, Factorial Products
• Index entries for sequences related to factorial numbers

Crossrefs

Cf. A034878, A001013, A075082, A085604, A115627, A322583.

Programs

PARI

A034876aux(n, m, p) = if(1==n, 1, my(s=0); forstep(i=m, p, -1, my(f=i!); if(!(n%f), s += A034876aux(n/f, i, 2))); (s));
A034876(n) = if(1==n,0,A034876aux(n!, n-1, precprime(n))); \\ (Slow) - Antti Karttunen, Dec 24 2018

PARI

A322583aux(n, m) = if(1==n, 1, my(s=0); for(i=2, oo, my(f=i!); if(f>m, return(s)); if(!(n%f), s += A322583aux(n/f, f))));
memoA322583 = Map();
A322583(n) = { my(c); if(mapisdefined(memoA322583,n,&c), c, c = A322583aux(n,n); mapput(memoA322583,n,c); (c)); };
A034876aux(n, m, p) = if(1==n, 1, my(s=0); forstep(i=m, p, -1, my(f=i!); s += A322583(n/f)); (s));
A034876(n) = if(1==n, 0, A034876aux(n!, n-1, precprime(n))); \\ Antti Karttunen, Dec 25 2018

Formula

a(1) = 0; for n > 1, a(n) = Sum_{x=A007917(n)..n-1} A322583(n!/x!) when n is a composite, and a(n) = 0 when n is a prime. - Antti Karttunen, Dec 25 2018

Extensions

Corrected by Jonathan Sondow, Dec 18 2004