A067080 If n = ab...def in decimal notation then the left digitorial function Ld(n) = ab...def*ab...de*ab...d*...*ab*a.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 90, 93, 96, 99, 102, 105, 108, 111, 114, 117, 160, 164, 168, 172, 176, 180, 184, 188, 192, 196, 250, 255, 260, 265, 270, 275, 280, 285, 290, 295, 360, 366, 372

Offset

1,2

Comments

This entry should probably start at n=0, just as A067079 does. But that would require a number of changes, so it can wait until the editors have more free time. - N. J. A. Sloane, Nov 29 2014

Links

Hieronymus Fischer, Table of n, a(n) for n = 1..10000

Formula

a(n) = Product_{k=1..length(n)} floor(n/10^(k-1)). - Vladeta Jovovic, Jan 08 2002

From Hieronymus Fischer, Aug 13 2007: (Start)

a(n) = product{0<=k<=floor(log_10(n)), floor(n/10^k)}, n>=1.

Recurrence:

a(n) = n*a(floor(n/10));

a(n*10^m) = n^m*10^(m(m+1)/2)*a(n).

a(k*10^m) = k^(m+1)*10^(m(m+1)/2), for 0<k<10.

a(n) <= b(n), where b(n)=n^(1+floor(log_10(n)))/10^(1/2*(1+floor(log_10(n)))*floor(log_10(n))); equality holds for n=k*10^m, m>=0, 1<=k<10. Here b(n) can also be written n^(1+floor(log_10(n)))/10^A000217(floor(log_10(n))).

Also: a(n) <= 3^((1-log_10(3))/2)*n^((1+log_10(n))/2)=1.332718...*10^A000217(log_10(n)), equality for n=3*10^m, m>=0.

a(n) > c*b(n), where c=0.472362443816572... (see constant A132026).

Also: a(n) > c*2^((1-log_10(2))/2)*n^((1+log_10(n))/2) = 0.601839...*10^A000217(log_10(n)).

lim inf a(n)/b(n) = 0.472362443816572..., for n-->oo.

lim sup a(n)/b(n) = 1, for n-->oo.

lim inf a(n)/n^((1+log_10(n))/2) = 0.472362443816572...*sqrt(2)/2^log_10(sqrt(2)), for n-->oo.

lim sup a(n)/n^((1+log_10(n))/2) = sqrt(3)/3^log_10(sqrt(3)), for n-->oo.

lim inf a(n)/a(n+1) = 0.472362443816572... for n-->oo (see constant A132026).

a(n) = O(n^((1+log_10(n))/2)). (End)

Example

Ld(256) = 256*25*2 =12800.
a(31)=floor(31/10^0)*floor(31/10^1)=31*3=93;
a(42)=168 since 42=42(base-10) and so a(42)=42*4(base-10)=42*4=168.

Mathematica

Table[d = IntegerDigits[n]; rd = 1; While[ Length[d] > 0, rd = rd*FromDigits[d]; d = Drop[d, -1]]; rd, {n, 1, 75} ]
Table[Times@@NestList[Quotient[#, 10]&, n, IntegerLength[n]-1], {n, 70}] (* Harvey P. Dale, Dec 16 2013 *)

Prog

(PARI) a(n)=my(t=n); while(n\=10, t*=n); t \\ Charles R Greathouse IV, Nov 20 2012
(Haskell)
a067080 n = if n <= 9 then n else n * a067080 (n `div` 10)
-- Reinhard Zumkeller, Nov 29 2012

Crossrefs

Cf. A067079, A065039, A048651, A098844, A132019, A132026, A132038, A000217.

For formulas regarding a general parameter p (i.e. terms floor(n/p^k)) see A132264.

For the product of terms floor(n/p^k) for p=2 to p=12 see A098844(p=2), A132027(p=3)-A132033(p=9), A132263(p=11), A132264(p=12).

For the products of terms 1+floor(n/p^k) see A132269-A132272, A132327, A132328.

Sequence in context: A341936 A066566 A256577 * A008554 A055644 A229621

Adjacent sequences: A067077 A067078 A067079 * A067081 A067082 A067083

Keyword

base,nonn,nice

Author

Amarnath Murthy, Jan 05 2002

Extensions

More terms from Robert G. Wilson v, Jan 07 2002

Status

approved