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 A034876 Number of ways to write n! as a product of smaller factorials each greater than 1. 6

%I

%S 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,

%T 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,

%U 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

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

%C 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

%C From _Antti Karttunen_, Dec 25 2018: (Start)

%C 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.

%C 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).

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

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

%C (End)

%D R. K. Guy, Unsolved Problems in Number Theory, B23.

%H Antti Karttunen, <a href="/A034876/b034876.txt">Table of n, a(n) for n = 1..69120</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FactorialProducts.html">Factorial Products</a>

%H <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>

%F 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

%e 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.

%e From _Antti Karttunen_, Dec 25 2018: (Start)

%e a(8) = 1 because 8! = 7! * (2!)^3.

%e a(9) = 1 because 9! = 7! * 3! * 3! * 2!.

%e a(16) = 2 because 16! = 15! * (2!)^4 = 14! * 5! * 2!.

%e a(144) = 2 because 144! = 143! * 4! * 3! = 143! * 3! * 3! * 2! * 2!.

%e a(576) = 3 because 576! = 575! * 4! * 4! = 575! * 4! * 3! * 2! * 2! = 575! * 3! * 3! * 2! * 2! * 2! * 2!.

%e a(720) = 2 because 720! = 719! * 6! = 719! * 5! * 3!.

%e a(3456) = 3 because 3456! = 3455! * 4! * 4! * 3! = 3455! * 4! * 3! * 3! * 2! * 2! = 3455! * 3! * 3! * 3! * 2! * 2! * 2! * 2!.

%e (End)

%o (PARI)

%o 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));

%o A034876(n) = if(1==n,0,A034876aux(n!, n-1, precprime(n))); \\ (Slow) - _Antti Karttunen_, Dec 24 2018

%o (PARI)

%o 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))));

%o memoA322583 = Map();

%o A322583(n) = { my(c); if(mapisdefined(memoA322583,n,&c), c, c = A322583aux(n,n); mapput(memoA322583,n,c); (c)); };

%o A034876aux(n, m, p) = if(1==n, 1, my(s=0); forstep(i=m, p, -1, my(f=i!); s += A322583(n/f)); (s));

%o A034876(n) = if(1==n, 0, A034876aux(n!, n-1, precprime(n))); \\ _Antti Karttunen_, Dec 25 2018

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

%K easy,nonn,nice

%O 1,10

%A _Erich Friedman_

%E Corrected by _Jonathan Sondow_, Dec 18 2004

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Last modified September 21 07:08 EDT 2019. Contains 327253 sequences. (Running on oeis4.)