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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
OFFSET
1,2
LINKS
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
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