OFFSET
1,2
COMMENTS
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).
FORMULA
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*(sqrt(2)/2^log_7(sqrt(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...*sqrt(2)/2^log_7(sqrt(2)), for n-->oo.
lim sup a(n)/n^((1+log_7(n))/2)=sqrt(3)/3^log_7(sqrt(3))=1.270209197..., for n-->oo.
lim inf a(n)/a(n+1)=0.4587667266997689850200... for n-->oo (see constant A132023).
EXAMPLE
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.
MATHEMATICA
Table[Times@@Floor[n/7^Range[0, Floor[Log[7, n]]]], {n, 70}] (* Harvey P. Dale, Oct 11 2017 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Hieronymus Fischer, Aug 20 2007
STATUS
approved