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Search: id:A132031
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| A132031 |
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Product{0<=k<=floor(log_7(n)), floor(n/7^k)}, n>=1. |
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3
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| 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 28, 30, 32, 34, 36, 38, 40, 63, 66, 69, 72, 75, 78, 81, 112, 116, 120, 124, 128, 132, 136, 175, 180, 185, 190, 195, 200, 205, 252, 258, 264, 270, 276, 282, 288, 343, 350, 357, 364, 371, 378, 385, 448, 456, 464, 472, 480, 488
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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If n is written in base-7 as n=d(m)d(m-1)d(m-2)...d(2)d(1)d(0) (where d(k) is the digit at position k) then a(n) is also the product d(m)d(m-1)d(m-2)...d(2)d(1)d(0)*d(m)d(m-1)d(m-2)...d(2)d(1)*d(m)d(m-1)d(m-2)...d(2)*...*d(m)d(m-1)d(m-2)*d(m)d(m-1)*d(m).
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FORMULA
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Recurrence: a(n)=n*a(floor(n/7)); a(n*7^m)=n^m*7^(m(m+1)/2)*a(n).
a(k*7^m)=k^(m+1)*7^(m(m+1)/2), for 0<k<7.
Asymptotic behavior: a(n)=O(n^((1+log_7(n))/2)this follows from the inequalities below.
a(n)<=b(n), where b(n)=n^(1+floor(log_7(n)))/7^((1+floor(log_7(n)))*floor(log_7(n))/2); equality holds for n=k*7^m, 0<k<7, m>=0. b(n) can also be written n^(1+floor(log_7(n)))/7^A000217(floor(log_7(n))).
Also: a(n)<=3^((1-log_7(3))/2)*n^((1+log_7(n))/2)=1.270209197...*7^A000217(log_7(n)), equality holds for n=3*7^m, m>=0.
a(n)>c*b(n), where c=0.4587667266997689850200... (see constant A132023).
Also: a(n)>c*(sqr(2)/2^log_7(sqr(2)))*n^((1+log_7(n))/2)=0.4587667266...*1.249972544...*7^A000217(log_7(n)).
lim inf a(n)/b(n)=0.4587667266997689850200..., for n-->oo.
lim sup a(n)/b(n)=1, for n-->oo.
lim inf a(n)/n^((1+log_7(n))/2)=0.4587667266997689850200...*sqr(2)/2^log_7(sqr(2)), for n-->oo.
lim sup a(n)/n^((1+log_7(n))/2)=sqr(3)/3^log_7(sqr(3))=1.270209197..., for n-->oo.
lim inf a(n)/a(n+1)=0.4587667266997689850200... for n-->oo (see constant A132023).
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EXAMPLE
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a(52)=floor(52/7^0)*floor(52/7^1)*floor(52/7^2)=52*7*1=364; a(58)=464 since 58=112(base-7) and so
a(58)=112*11*1(base-7)=58*8*1=464.
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CROSSREFS
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Cf. A048651, A132023, A132035, 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), A067080(p=10), A132263(p=11), A132264(p=12).
For the products of terms 1+floor(n/p^k) see A132269-A132272, A132327, A132328.
Sequence in context: A023767 A023794 A032948 this_sequence A072763 A057846 A055647
Adjacent sequences: A132028 A132029 A132030 this_sequence A132032 A132033 A132034
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KEYWORD
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nonn
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AUTHOR
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Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 20 2007
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