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A131451 Product of the nonzero digital products of all the numbers 1 to n (a 'total digital-product factorial' in base 10). 22
1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 362880, 362880, 725760, 2177280, 8709120, 43545600, 261273600, 1828915200, 14631321600, 131681894400, 263363788800, 526727577600, 2106910310400, 12641461862400, 101131694899200 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

Hieronymus Fischer, Table of n, a(n) for n = 1..102

FORMULA

The following formulas are given for general bases p>1:

a(n)=product{1<=k<=n, dp_p(k)} where dp_p(k) = product of the nonzero digits of k in base p.

a(n)=(n mod p)!*product{0<j<=log_p(n)}(p-1)!^(floor(n/p^j)*p^(j-1)) * product{0<j<=log_p(n),(floor(n/p^j)mod p)>0}(floor(n/p^j)mod p)^(1+(n mod p^j))*((floor(n/p^j)mod p)-1)!^(p^j).

Recurrence: a(n+k*p^m)=a(n)*k^n*a(k*p^m) for 0<=k<p, 0<=n<p^m.

a(n)=n!, for 0<=n<p.

a(k*p^m)=k*(p-1)!^(k*m*p^(m-1))*(k-1)!^(p^m) for 0<=k<p.

a(n)=(p-1)!^((m*p^(m+1)-(m+1)*p^m+1)/(p-1)^2)=(p-1)!^(1+2*p+3*p^2+...+m*p^(m-1)) for n=1+p+p^2+...+p^m.

a(n)=(p-1)!^(k*(m*p^(m+1)-(m+1)*p^m+1)/(p-1)^2)*(k-1)!^(p*(p^m-1)/(p-1))*k^(k*(p^(m+1)-(m+1)*p+m)/(p-1)^2)*k!*k^m, for n=k*(1+p+p^2+...+p^m).

For p=10: a(10^n)=9!^(n*10^(n-1)).

Asymptotic behavior: a(10^n)=10^(0.5559763...*n*10^n). Hence it grows slower than the factorial A000142(10^n) for which we have (10^n)!=10^((n-0.43429448...)*10^n+n/2+0.3990899...+o(1/n)). Example: a(1000) has 1668 digits, whereas 1000! has 2568 digits.

EXAMPLE

a(12)=dp_10(1)*dp_10(2)*dp_10(3)*...*dp_10(11)*dp_10(12)=1*2*3*4*5*6* 7*8*9*1*(1*1)*(1*2).

a(345)=3*4*5*3^45*4^5*(3-1)!^100*(4-1)!^10*(5-1)!^1*9!^64.

a(1000)=9!^300. a(1111)=9!^321.

MAPLE

with transforms;

f:=proc(n) option remember; if n = 0 then 1 else f(n-1)*digprod0(n); fi; end; [seq(f(n), n=0..40)]; # N. J. A. Sloane, Oct 12 2013

CROSSREFS

Cf. A131385, A010786, A131387, A007953, A007954, A037131.

Sequence in context: A248778 A033647 A109834 * A084012 A062919 A276843

Adjacent sequences:  A131448 A131449 A131450 * A131452 A131453 A131454

KEYWORD

nonn,base

AUTHOR

Hieronymus Fischer, Jul 11 2007

EXTENSIONS

New b-file from Hieronymus Fischer, Sep 10 2007

2 typos in the formula section removed by Hieronymus Fischer, Dec 05 2011

STATUS

approved

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Last modified June 28 16:56 EDT 2017. Contains 288839 sequences.