This site is supported by donations to The OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A132264 Product{0<=k<=floor(log_12(n)), floor(n/12^k)}, n>=1. 16
 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 108, 111, 114, 117, 120, 123, 126, 129, 132, 135, 138, 141, 192, 196, 200, 204, 208, 212, 216, 220, 224, 228, 232, 236, 300, 305, 310, 315 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS If n is written in base-12 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). LINKS FORMULA The following formulas are given for a general parameter p considering the product of terms floor(n/p^k) for 0<=k<=floor(log_p(n)), where p=12 for this sequence. Recurrence: a(n)=n*a(floor(n/p)); a(n*p^m)=n^m*p^(m(m+1)/2)*a(n). a(k*p^m)=k^(m+1)*p^(m(m+1)/2), for 0=0. b(n) can also be written n^(1+floor(log_p(n)))/p^A000217(floor(log_p(n))). Also: a(n)<=q^((1-log_p(q))/2)*n^((1+log_p(n))/2)=q^((1-log_p(q))/2)*p^A000217(log_p(n)), equality holds for n=q*p^m, m>=0, where q=floor(sqrt(p)+1/2). Also, equality holds for n=(q+1)*p^m, provided p is a A002378-number (in this case we have p=q*(q+1) and so q^((1-log_p(q))/2)=(q+1)^((1-log_p(q+1))/2)). a(n)>c*b(n), where c=product{k>0, 1-1/(2*p^k)}=0.47735217025489380... (for p=12 see constant A132265). Also: a(n)>c*(sqr(2)/2^log_p(sqr(2)))*n^((1+log_p(n))/2)=0.612870619...*p^A000217(log_p(n)), (p=12). lim inf a(n)/b(n)=product{k>0, 1-1/(2*p^k)}=0.47735217025489380198334286365820..., for n-->oo (for p=12 see constant A132265). lim sup a(n)/b(n)=1, for n-->oo. lim inf a(n)/n^((1+log_p(n))/2)=(sqr(2)/2^log_p(sqr(2)))*product{k>0, 1-1/(2*p^k)}=0.612870619..., for n-->oo, (p=12). lim sup a(n)/n^((1+log_p(n))/2)=sqr(q)/q^log_p(sqr(q))=1.358593737..., for n-->oo, (p=12, q=round(sqrt(p))=3). lim inf a(n)/a(n+1)=product{k>0, 1-1/(2*p^k)}=0.47735217025489380... for n-->oo (for p=12 see constant A132265). EXAMPLE a(50)=floor(50/12^0)*floor(50/12^1)=50*4=200; a(65)=325 since 65=55(base-12) and so a(65)=55*5(base-12)=65*5=325. CROSSREFS Cf. A048651, A132019-A132026, A132265, A132267, A000217, A002378. For the product of terms floor(n/p^k) for p=2 to p=11 see A098844(p=2), A132027(p=3)-A132033(p=9), A067080(p=10), A132263(p=11). For the products of terms 1+floor(n/p^k) see A132269-A132272, A132327, A132328. Sequence in context: A135382 A064598 A048242 * A214922 A004830 A081330 Adjacent sequences:  A132261 A132262 A132263 * A132265 A132266 A132267 KEYWORD nonn,base AUTHOR Hieronymus Fischer, Aug 20 2007 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Puzzles | Hot | Classics
Recent Additions | More pages | Superseeker | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .