A132029 Product{0<=k<=floor(log_5(n)), floor(n/5^k)}, n>=1.

1, 2, 3, 4, 5, 6, 7, 8, 9, 20, 22, 24, 26, 28, 45, 48, 51, 54, 57, 80, 84, 88, 92, 96, 125, 130, 135, 140, 145, 180, 186, 192, 198, 204, 245, 252, 259, 266, 273, 320, 328, 336, 344, 352, 405, 414, 423, 432, 441, 1000, 1020, 1040, 1060, 1080, 1210, 1232, 1254, 1276

Offset

1,2

Comments

If n is written in base-5 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

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Formula

Recurrence: a(n)=n*a(floor(n/5)); a(n*5^m)=n^m*5^(m(m+1)/2)*a(n).

a(k*5^m)=k^(m+1)*5^(m(m+1)/2), for 0<k<5.

Asymptotic behavior: a(n)=O(n^((1+log_5(n))/2)); this follows from the inequalities below.

a(n)<=b(n), where b(n)=n^(1+floor(log_5(n)))/5^((1+floor(log_5(n)))*floor(log_5(n))/2); equality holds for n=k*5^m, 0<k<5, m>=0. b(n) can also be written n^(1+floor(log_5(n)))/5^A000217(floor(log_5(n))).

Also: a(n)<=2^((1-log_5(2))/2)*n^((1+log_5(n))/2)=1.2181246...*5^A000217(log_5(n)), equality holds for n=2*5^m, m>=0.

a(n)>c*b(n), where c=0.438796837203638531... (see constant A132021).

Also: a(n)>c*(sqrt(2)/2^log_5(sqrt(2)))*n^((1+log_5(n))/2)=0.534509224...*5^A000217(log_5(n)).

lim inf a(n)/b(n)=0.438796837203638531..., for n-->oo.

lim sup a(n)/b(n)=1, for n-->oo.

lim inf a(n)/n^((1+log_5(n))/2)=0.438796837203638531...*sqrt(2)/2^log_5(sqrt(2)), for n-->oo.

lim sup a(n)/n^((1+log_5(n))/2)=sqrt(2)/2^log_5(sqrt(2))=1.2181246..., for n-->oo.

lim inf a(n)/a(n+1)=0.438796837203638531... for n-->oo (see constant A132021).

Example

a(26)=floor(26/5^0)*floor(26/5^1)*floor(26/5^2)=26*5*1=130; a(34)=204 since 34=114(base-5) and so a(34)=114*11*1(base-5)=34*6*1=204.

Mathematica

Table[Product[Floor[n/5^k], {k, 0, Floor[Log[5, n]]}], {n, 60}] (* Harvey P. Dale, Oct 16 2019 *)

Crossrefs

Cf. A048651, A132021, A100222, 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: A109795 A248500 A250041 * A281724 A217489 A217491

Adjacent sequences: A132026 A132027 A132028 * A132030 A132031 A132032

Keyword

nonn

Author

Hieronymus Fischer, Aug 20 2007

Status

approved