A132031 Product{0<=k<=floor(log_7(n)), floor(n/7^k)}, n>=1.

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, 81, 112, 116, 120, 124, 128, 132, 136, 175, 180, 185, 190, 195, 200, 205, 252, 258, 264, 270, 276, 282, 288, 343, 350, 357, 364, 371, 378, 385, 448, 456, 464, 472, 480, 488

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).

Links

Table of n, a(n) for n=1..61.

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

Cf. A048651, A132023, A132035, 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: A023794 A032948 A377915 * A323175 A323204 A072763

Adjacent sequences: A132028 A132029 A132030 * A132032 A132033 A132034

Keyword

nonn

Author

Hieronymus Fischer, Aug 20 2007

Status

approved