0,2

If n is written in base-2 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)d(0))*(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)).

Recurrence: a(n)=(1+n)*a(floor(n/2)); a(2n)=(1+2n)*a(n); a(n*2^m)=product{1<=k<=m, 1+n*2^k}*a(n).

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

a(n)=A132270(2n)=(1+n)*A132270(n).

Asymptotic behavior: a(n)=O(n^((1+log_2(n))/2)); this follows from the inequalities below.

a(n)<=A098844(n)*product{0<=k<=floor(log_2(n)), 1+1/2^k}.

a(n)>=A098844(n)/product{1<=k<=floor(log_2(n)), 1-1/2^k}.

a(n)<c*n^((1+log_2(n))/2)=c*2^A000217(log_2(n)), where c=product{k>=0, 1+1/2^k}=4.7684620580627... (see constant A081845).

a(n)>n^((1+log_2(n))/2)=2^A000217(log_2(n)),

lim sup a(n)/A098844(n)=product{k>=0, 1+1/2^k}=4.7684620580627..., for n-->oo (see constant A081845).

lim inf a(n)/A098844(n)=1/product{k>0, 1-1/2^k}=1/0.288788095086602421..., for n-->oo (see constant A048651).

lim inf a(n)/n^((1+log_2(n))/2)=1, for n-->oo.

lim sup a(n)/n^((1+log_2(n))/2)=product{k>=0, 1+1/2^k}=4.7684620580627..., for n-->oo (see constant A081845).

lim inf a(n+1)/a(n)=product{k>=0, 1+1/2^k}=4.7684620580627... for n-->oo (see constant A081845).

a(10)=(1+floor(10/2^0))*(1+floor(10/2^1))*(1+floor(10/2^2))*(1+floor(10/2^3))=11*6*3*2=396; a(17)=4860 since

17=10001(base-2) and so a(17)=(1+10001)*(1+1000)*(1+100)*(1+10)*(1+1)(base-2)=18*9*5*3*2=4860.

Cf. A048651, A081845, A132270, A132271(p=10), A132272, A132327(p=3), A132328.

For formulas regarding a general parameter p (i.e. terms 1+floor(n/p^k)) see A132271.

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

Adjacent sequences: A132266 A132267 A132268 this_sequence A132270 A132271 A132272

Sequence in context: A057852 A056188 A020696 this_sequence A053287 A086323 A075999

nonn

Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 20 2007