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A034876 Number of ways to write n! as a product of smaller factorials each greater than 1. 6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,10

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)

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

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

EXAMPLE

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)

PROG

(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

CROSSREFS

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

Sequence in context: A219558 A279210 A232243 * A091393 A284557 A182032

Adjacent sequences:  A034873 A034874 A034875 * A034877 A034878 A034879

KEYWORD

easy,nonn,nice

AUTHOR

Erich Friedman

EXTENSIONS

Corrected by Jonathan Sondow, Dec 18 2004

STATUS

approved

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Last modified August 19 04:18 EDT 2019. Contains 326109 sequences. (Running on oeis4.)