cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

A131451 Product of the nonzero digital products of all the numbers 1 to n (a 'total digital-product factorial' in base 10).

Original entry on oeis.org

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

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Author

Hieronymus Fischer, Jul 11 2007

Keywords

Examples

			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.
		

Crossrefs

Programs

  • 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

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{00}(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
a(n)=n!, for 0<=n
a(k*p^m)=k*(p-1)!^(k*m*p^(m-1))*(k-1)!^(p^m) for 0<=k
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.

Extensions

New b-file from Hieronymus Fischer, Sep 10 2007
2 typos in the formula section removed by Hieronymus Fischer, Dec 05 2011