A067080 - OEIS

%I #34 Feb 18 2024 08:19:02

%S 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,

%T 54,56,58,90,93,96,99,102,105,108,111,114,117,160,164,168,172,176,180,

%U 184,188,192,196,250,255,260,265,270,275,280,285,290,295,360,366,372

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

%C 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

%H Hieronymus Fischer, <a href="/A067080/b067080.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = Product_{k=1..length(n)} floor(n/10^(k-1)). - _Vladeta Jovovic_, Jan 08 2002

%F From _Hieronymus Fischer_, Aug 13 2007: (Start)

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

%F Recurrence:

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

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

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

%F 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))).

%F 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.

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

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

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

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

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

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

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

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

%e Ld(256) = 256*25*2 =12800.

%e a(31)=floor(31/10^0)*floor(31/10^1)=31*3=93;

%e a(42)=168 since 42=42(base-10) and so a(42)=42*4(base-10)=42*4=168.

%t Table[d = IntegerDigits[n]; rd = 1; While[ Length[d] > 0, rd = rd*FromDigits[d]; d = Drop[d, -1]]; rd, {n, 1, 75} ]

%t Table[Times@@NestList[Quotient[#,10]&,n,IntegerLength[n]-1],{n,70}] (* _Harvey P. Dale_, Dec 16 2013 *)

%o (PARI) a(n)=my(t=n);while(n\=10,t*=n); t \\ _Charles R Greathouse IV_, Nov 20 2012

%o (Haskell)

%o a067080 n = if n <= 9 then n else n * a067080 (n `div` 10)

%o -- _Reinhard Zumkeller_, Nov 29 2012

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

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

%Y 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).

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

%K base,nonn,nice

%O 1,2

%A _Amarnath Murthy_, Jan 05 2002

%E More terms from _Robert G. Wilson v_, Jan 07 2002