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%I #11 Oct 11 2017 16:59:00
%S 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,
%T 81,112,116,120,124,128,132,136,175,180,185,190,195,200,205,252,258,
%U 264,270,276,282,288,343,350,357,364,371,378,385,448,456,464,472,480,488
%N Product{0<=k<=floor(log_7(n)), floor(n/7^k)}, n>=1.
%C 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).
%F Recurrence: a(n)=n*a(floor(n/7)); a(n*7^m)=n^m*7^(m(m+1)/2)*a(n).
%F a(k*7^m)=k^(m+1)*7^(m(m+1)/2), for 0<k<7.
%F Asymptotic behavior: a(n)=O(n^((1+log_7(n))/2)this follows from the inequalities below.
%F 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))).
%F 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.
%F a(n)>c*b(n), where c=0.4587667266997689850200... (see constant A132023).
%F Also: a(n)>c*(sqrt(2)/2^log_7(sqrt(2)))*n^((1+log_7(n))/2)=0.4587667266...*1.249972544...*7^A000217(log_7(n)).
%F lim inf a(n)/b(n)=0.4587667266997689850200..., for n-->oo.
%F lim sup a(n)/b(n)=1, for n-->oo.
%F lim inf a(n)/n^((1+log_7(n))/2)=0.4587667266997689850200...*sqrt(2)/2^log_7(sqrt(2)), for n-->oo.
%F lim sup a(n)/n^((1+log_7(n))/2)=sqrt(3)/3^log_7(sqrt(3))=1.270209197..., for n-->oo.
%F lim inf a(n)/a(n+1)=0.4587667266997689850200... for n-->oo (see constant A132023).
%e a(52)=floor(52/7^0)*floor(52/7^1)*floor(52/7^2)=52*7*1=364.
%e a(58)=464 since 58=112(base-7) and so a(58)=112*11*1(base-7)=58*8*1=464.
%t Table[Times@@Floor[n/7^Range[0,Floor[Log[7,n]]]],{n,70}] (* _Harvey P. Dale_, Oct 11 2017 *)
%Y Cf. A048651, A132023, A132035, A000217.
%Y For formulas regarding a general parameter p (i.e. terms floor(n/p^k)) see A132264.
%Y 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).
%Y For the products of terms 1+floor(n/p^k) see A132269-A132272, A132327, A132328.
%K nonn
%O 1,2
%A _Hieronymus Fischer_, Aug 20 2007
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