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Search: id:A132030
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| A132030 |
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Product{0<=k<=floor(log_6(n)), floor(n/6^k)}, n>=1. |
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4
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| 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 24, 26, 28, 30, 32, 34, 54, 57, 60, 63, 66, 69, 96, 100, 104, 108, 112, 116, 150, 155, 160, 165, 170, 175, 216, 222, 228, 234, 240, 246, 294, 301, 308, 315, 322, 329, 384, 392, 400, 408, 416, 424, 486, 495, 504, 513, 522, 531, 600
(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-6 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/6)); a(n*6^m)=n^m*6^(m(m+1)/2)*a(n).
a(k*6^m)=k^(m+1)*6^(m(m+1)/2), for 0<k<6.
Asymptotic behavior: a(n)=O(n^((1+log_6(n))/2)); this follows from the inequalities below.
a(n)<=b(n), where b(n)=n^(1+floor(log_6(n)))/6^((1+floor(log_6(n)))*floor(log_6(n))/2); equality holds for n=k*6^m, 0<k<6, m>=0. b(n) can also be written n^(1+floor(log_6(n)))/6^A000217(floor(log_6(n))).
Also: a(n)<=2^((1-log_6(2))/2)*n^((1+log_6(n))/2)=1.236766885...*6^A000217(log_6(n)), equality holds for n=2*6^m and for n=3*6^m, m>=0 (consider 2^((1-log_6(2))/2)=3^((1-log_6(3))/2) since 6=2*3).
a(n)>c*b(n), where c=0.45071262522603913... (see constant A132022).
Also: a(n)>c*(sqr(2)/2^log_6(sqr(2)))*n^((1+log_6(n))/2)=0.557426449...*6^A000217(log_6(n)).
lim inf a(n)/b(n)=0.45071262522603913..., for n-->oo.
lim sup a(n)/b(n)=1, for n-->oo.
lim inf a(n)/n^((1+log_6(n))/2)=0.45071262522603913...*sqr(2)/2^log_6(sqr(2)), for n-->oo.
lim sup a(n)/n^((1+log_6(n))/2)=sqr(3)/3^log_6(sqr(3))=1.236766885..., for n-->oo.
lim inf a(n)/a(n+1)=0.45071262522603913... for n-->oo (see constant A132022).
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EXAMPLE
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a(52)=floor(52/6^0)*floor(52/6^1)*floor(52/6^2)=52*8*1=416; a(58)=522 since 58=134(base-6) and so
a(58)=134*13*1(base-6)=58*9*1=522.
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CROSSREFS
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Cf. A048651, A132022, A132034, 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: A023792 A032946 A039156 this_sequence A004851 A066638 A062279
Adjacent sequences: A132027 A132028 A132029 this_sequence A132031 A132032 A132033
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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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