This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A132327 #9 Mar 07 2025 21:30:22
%S A132327 1,2,3,8,10,12,21,24,27,80,88,96,130,140,150,192,204,216,399,420,441,
%T A132327 528,552,576,675,702,729,2240,2320,2400,2728,2816,2904,3264,3360,3456,
%U A132327 4810,4940,5070,5600,5740,5880,6450,6600,6750,8832,9024,9216,9996,10200
%N A132327 a(n) = Product{k>=0} (1 + floor(n/3^k)).
%C A132327 If n is written in base-3 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)).
%F A132327 Recurrence: a(n)=(1+n)*a(floor(n/3)); a(3n)=(1+3n)*a(n); a(n*3^m)=product{1<=k<=m, 1+n*3^k}*a(n).
%F A132327 a(k*3^m-j)=(k*3^m-j+1)*3^m*p^(m(m-1)/2), for 0<k<3, 0<j<3, m>=1, a(3^m)=3^(m(m+1)/2)*product{0<=k<=m, 1+1/3^k}, m>=1.
%F A132327 a(n)=A132328(3*n)=(1+n)*A132328(n).
%F A132327 Asymptotic behavior: a(n)=O(n^((1+log_3(n))/2)); this follows from the inequalities below.
%F A132327 a(n)<=A132027(n)*product{0<=k<=floor(log_3(n)), 1+1/3^k}.
%F A132327 a(n)>=A132027(n)/product{1<=k<=floor(log_3(n)), 1-1/3^k}.
%F A132327 a(n)<c*n^((1+log_3(n))/2)=c*3^A000217(log_3(n)), where c=product{k>=0, 1+1/p^k}=3.12986803713402307587769821345767... (see constant A132323).
%F A132327 a(n)>n^((1+log_3(n))/2)=3^A000217(log_3(n)).
%F A132327 lim sup a(n)/A132027(n)=2*product{k>0, 1+1/3^k}=3.12986803713402307587769821345767..., for n-->oo (see constant A132323).
%F A132327 lim inf a(n)/A132027(n)=1/product{k>0, 1-1/3^k}=1/0.560126077927948944969792243314140014..., for n-->oo (see constant A100220).
%F A132327 lim inf a(n)/n^((1+log_3(n))/2)=1, for n-->oo.
%F A132327 lim sup a(n)/n^((1+log_3(n))/2)=2*product{k>0, 1+1/3^k}=3.12986803713402307587769821345767..., for n-->oo (see constant A132323).
%F A132327 lim inf a(n+1)/a(n)=2*product{k>0, 1+1/3^k}=3.12986803713402307587769821345767... for n-->oo (see constant A132323).
%e A132327 a(12)=(1+floor(12/3^0))*(1+floor(12/3^1))*(1+floor(12/3^2))=13*5*2=130; a(20)=441 since 20=202(base-3) and so
%e A132327 a(20)=(1+202)*(1+20)*(1+2)(base-3)=21*7*3=441.
%t A132327 Table[Product[1+Floor[n/3^k],{k,0,n}],{n,0,49}] (* _James C. McMahon_, Mar 07 2025 *)
%Y A132327 Cf. A100220, A132027, A132038, A132264, A132269(for p=2), A132271(for p=10).
%Y A132327 For formulas regarding a general parameter p (i.e. terms 1+floor(n/p^k)) see A132271.
%Y A132327 For the product of terms floor(n/p^k) see A098844, A067080, A132027-A132033, A132263, A132264.
%K A132327 nonn,base
%O A132327 0,2
%A A132327 _Hieronymus Fischer_, Aug 20 2007