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Search: id:A098844
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| A098844 |
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a(1)=1, a(n)=n*a(floor(n/2)). |
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+0
45
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| 1, 2, 3, 8, 10, 18, 21, 64, 72, 100, 110, 216, 234, 294, 315, 1024, 1088, 1296, 1368, 2000, 2100, 2420, 2530, 5184, 5400, 6084, 6318, 8232, 8526, 9450, 9765, 32768, 33792, 36992, 38080, 46656, 47952, 51984, 53352, 80000, 82000, 88200, 90300
(list; graph; listen)
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OFFSET
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1,2
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FORMULA
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a(2^n)=2^(n*(n+1)/2)=A006125(n+1)
a(n)=product{0<=k<=floor(log_2(n)), floor(n/2^k)}, n>=1. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 13 2007
Recurrence: a(n*2^m)=n^m*2^(m(m+1)/2)*a(n). - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 13 2007
a(n)<=n^((1+log_2(n))/2)=2^A000217(log_2(n)); equality iff n is a power of 2. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 13 2007
a(n)>=c(n)*(n+1)^((1+log_2(n+1))/2) for n<>2, where c(n)=product{1<=k<=floor(log_2(n)), 1-1/2^k}; equality iff n+1 is a power of 2. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 13 2007
a(n)>c*(n+1)^((1+log_2(n+1))/2) where c=0.288788095086602421... (see constant A048651). - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 13 2007
lim inf a(n)/n^((1+log_2(n))/2)=0.288788095086602421... for n-->oo. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 13 2007
lim sup a(n)/n^((1+log_2(n))/2)=1 for n-->oo. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 13 2007
lim inf a(n)/a(n+1)=0.288788095086602421... for n-->oo (see constant A048651). - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 13 2007
a(n)=O(n^((1+log_2(n))/2)). - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 20 2007
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EXAMPLE
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a(10)=floor(10/2^0)*floor(10/2^1)*floor(10/2^2)*floor(10/2^3)=10*5*2*1=100;
a(17)=1088 since 17=10001(base-2) and so a(17)=10001*1000*100*10*1(base-2)=17*8*4*2*1=1088.
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PROGRAM
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(PARI) a(n)=if(n<2, 1, n*a(floor(n/2)))
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CROSSREFS
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Cf. A048651, A067080, A132027, A132028, A132029, A132030, A132019, A132026, A132038.
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=3 to p=12 see 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: A060697 A093353 A083799 this_sequence A034437 A138880 A063474
Adjacent sequences: A098841 A098842 A098843 this_sequence A098845 A098846 A098847
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KEYWORD
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nonn
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AUTHOR
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Benoit Cloitre (benoit7848c(AT)orange.fr), Nov 03 2004
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