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 A123709 #30 Jan 25 2021 19:05:52
%S A123709 1,2,3,4,3,4,3,4,4,6,3,8,3,6,7,4,3,8,3,8,7,6,3,8,4,6,4,8,3,11,3,4,7,6,
%T A123709 7,8,3,6,7,8,3,11,3,8,8,6,3,8,4,8,7,8,3,8,7,8,7,6,3,16,3,6,8,4,7,12,3,
%U A123709 8,7,14,3,8,3,6,8,8
%N A123709 a(n) is the number of nonzero elements in row n of triangle A123706.
%C A123709 Triangle A123706 is the matrix inverse of triangle A010766, where A010766(n,k) = [n/k]. a(n) = 4 when n is in A123710. a(n) = 8 when n is in A123711. a(n) = 16 when n is in A123712.
%H A123709 M. F. Hasler, <a href="/A123709/b123709.txt">Table of n, a(n) for n = 1..500</a>
%H A123709 Peter Luschny, <a href="http://groups.google.com/group/seqcomp/msg/2aabf27ef9b698ef">Re: Is the A123706 triangle an extension of the Moebius function?</a>, seqcomp list, Feb 12 2012
%F A123709 a(n) = 2^(m+1) - 1 when n is the product of m distinct odd primes. [Corrected by _M. F. Hasler_, Feb 13 2012]
%F A123709 For any k>1, a(n)=2^k if, and only if, n is a nonsquarefree number with A001221(n) = k-1 (= omega(n), number of distinct prime factors), with the only exception of a(n=6)=2^2. - _M. F. Hasler_, Feb 12 2012
%F A123709 A123709(n) = 1 + #{ k in 1..n-1 | Moebius(n,k+1) <> Moebius(n,k) }, where Moebius(n,k)={moebius(n/k) if n=0 (mod k), 0 else}, cf. link to message by P. Luschny. - _M. F. Hasler_, Feb 13 2012
%e A123709 a(n) = 3 when n is an odd prime.
%e A123709 a(n) = 7 when n is the product of two different odd primes. [Corrected by _M. F. Hasler_, Feb 13 2012]
%e A123709 a(n) = 15 when n is the product of three different odd primes. [Corrected by _M. F. Hasler_, Feb 13 2012]
%t A123709 Moebius[i_,j_]:=If[Divisible[i,j], MoebiusMu[i/j],0];
%t A123709 A123709[n_]:=Length[Select[Table[Moebius[n,j]-Moebius[n,j+1],{j,1,n}],#!=0&]];
%t A123709 Array[A123709, 500] (* _Enrique Pérez Herrero_, Feb 13 2012 *)
%o A123709 (PARI) {a(n)=local(M=matrix(n,n,r,c,if(r>=c,floor(r/c)))^-1); sum(k=1,n,if(M[n,k]==0,0,1))}
%o A123709 (PARI) A123709(n)=#select((matrix(n, n, r, c, r\c)^-1)[n,],x->x) \\ _M. F. Hasler_, Feb 12 2012
%o A123709 (PARI) A123709(n)={ my(t=moebius(n)); sum(k=2,n, t+0 != t=if(n%k,0,moebius(n\k)))+1} /* the "t+0 != ..." is required because of a bug in PARI versions <= 2.4.2, maybe beyond, which seems to be fixed in v. 2.5.1 */ \\ _M. F. Hasler_, Feb 13 2012
%Y A123709 Cf. A123706, A123707, A123708, A123710, A123711, A123712; A010766.
%K A123709 nonn
%O A123709 1,2
%A A123709 _Paul D. Hanna_, Oct 09 2006