A132272 Product{k>0, 1+floor(n/10^k)}.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10

Offset

0,11

Comments

If n is written in base-10 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 (1+d(m)d(m-1)d(m-2)...d(2)d(1))*(1+d(m)d(m-1)d(m-2)...d(2))*...*(1+d(m)d(m-1)d(m-2))*(1+d(m)d(m-1))*(1+d(m)).

a(n) = A179051(n) for n < 100. [From Reinhard Zumkeller, Jun 27 2010]

Links

Table of n, a(n) for n=0..99.

Formula

The following formulas are given for a general parameter p considering the product of terms 1+floor(n/p^k) for 0<k<=floor(log_p(n)), where p=10 for this sequence.

Recurrence: a(n)=(1+floor(n/p))*a(floor(n/p)); a(pn)=(1+n)*a(n); a(n*p^m)=product{0<=k<m, 1+n*p^k}*a(n).

a(k*p^m-j)=k^m*p^(m(m-1)/2), for 0<k<p, 0<j<p, m>=1. a(p^m)=p^(m(m-1)/2)*product{0<=k<m, 1+1/p^k}

a(n)=A132271(floor(n/p)=A132271(n)/(1+n).

Asymptotic behavior: a(n)=O(n^((log_p(n)-1)/p)); this follows from the inequalities below.

a(n)<=A067080(n)/(n+1)*product{0<=k<=floor(log_p(n)), 1+1/p^k}.

a(n)>=A067080(n)/((n+1)*product{0<k<=floor(log_p(n)), 1-1/p^k}).

a(n)<c*n^((1+log_p(n))/2)/(n+1)=c*2^A000217(log_p(n))/(n+1), where c=product{k>0, 1+1/p^k}=2.2244691382741012... (for p=10 see constant A132325).

a(n)>n^((1+log_p(n))/2)/(n+1)=p^A000217(log_p(n))/(n+1).

lim sup n*a(n)/A067080(n)=2*product{k>0, 1+1/p^k}=2.2244691382741012..., for n-->oo (for p=10 see constant A132325).

lim inf n*a(n)/A067080(n)=1/product{k>0, 1-1/p^k}=1/0.8900100999989990000001000..., for n-->oo (for p=10 s. constant A132038).

lim inf a(n)/n^((1+log_p(n))/2)=1, for n-->oo.

lim sup a(n)/n^((1+log_p(n))/2)=2*product{k>0, 1+1/p^k}=2.2244691382741012..., for n-->oo (for p=10 see constant A132325).

lim inf a(n+1)/a(n)=2*product{k>0, 1+1/p^k}=2.2244691382741012... for n-->oo (for p=10 see constant A132325).

Example

a(121)=(1+floor(121/10^1))*(1+floor(121/10^2))=13*2=26; a(132)=28 since 132=132(base-10) and so
a(132)=(1+13)*(1+1)(base-10)=14*2=28.

Crossrefs

Cf. A132038, A132270(p=2), A132271, A132325, A132328(p=3), A132271.

For the product of terms floor(n/p^k) see A098844, A067080, A132027-A132033, A132263, A132264.

Sequence in context: A226233 A329624 A059995 * A179051 A324160 A054899

Adjacent sequences: A132269 A132270 A132271 * A132273 A132274 A132275

Keyword

nonn,base

Author

Hieronymus Fischer, Aug 20 2007

Status

approved