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A067080
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If n = ab...def in decimal notation then the left digitorial function Ld(n) = ab...def*ab...de*ab...d*...*ab*a.
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48
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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FORMULA
| Product_{k=1..length(n)} floor(n/10^(k-1)). - Vladeta Jovovic (vladeta(AT)eunet.rs), Jan 08 2002
a(n)=product{0<=k<=floor(log_10(n)), floor(n/10^k)}, n>=1. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 13 2007
Recurrence: a(n)=n*a(floor(n/10)); a(n*10^m)=n^m*10^(m(m+1)/2)*a(n). - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 13 2007
a(k*10^m)=k^(m+1)*10^(m(m+1)/2), for 0<k<10. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 13 2007
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))). - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 13 2007
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. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 13 2007
a(n)>c*b(n), where c=0.472362443816572... (see constant A132026). - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 13 2007
Also: a(n)>c*2^((1-log_10(2))/2)*n^((1+log_10(n))/2)=0.601839...*10^A000217(log_10(n)). - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 13 2007
lim inf a(n)/b(n)=0.472362443816572..., for n-->oo. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 13 2007
lim sup a(n)/b(n)=1, for n-->oo. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 13 2007
lim inf a(n)/n^((1+log_10(n))/2)=0.472362443816572...*sqr(2)/2^log_10(sqr(2)), for n-->oo. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 13 2007
lim sup a(n)/n^((1+log_10(n))/2)=sqr(3)/3^log_10(sqr(3)), for n-->oo. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 13 2007
lim inf a(n)/a(n+1)=0.472362443816572... for n-->oo (see constant A132026). - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 13 2007
a(n)=O(n^((1+log_10(n))/2)). - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 20 2007
3 typos or coding errors, respectively, removed by Hieronymus Fischer
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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.
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MATHEMATICA
| Table[d = IntegerDigits[n]; rd = 1; While[ Length[d] > 0, rd = rd*FromDigits[d]; d = Drop[d, -1]]; rd, {n, 1, 75} ]
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CROSSREFS
| Cf. A067079, A065039.
Cf. 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: A057605 A110548 A066566 * A008554 A055644 A077342
Adjacent sequences: A067077 A067078 A067079 * A067081 A067082 A067083
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KEYWORD
| base,nonn,nice
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AUTHOR
| Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Jan 05 2002
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EXTENSIONS
| More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Jan 07 2002
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