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A227291 Characteristic function of squarefree numbers squared (A062503).

%I A227291 #50 Nov 27 2024 15:04:43
%S A227291 1,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,
%T A227291 0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%U A227291 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0
%N A227291 Characteristic function of squarefree numbers squared (A062503).
%H A227291 Antti Karttunen, <a href="/A227291/b227291.txt">Table of n, a(n) for n = 1..100000</a> (first 10000 terms from Reinhard Zumkeller)
%H A227291 <a href="/index/Ch#char_fns">Index entries for characteristic functions</a>
%H A227291 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%F A227291 Dirichlet g.f.: zeta(2s)/zeta(4s) = prod[prime p: 1+p^(-2s) ], see A008966.
%F A227291 a(n) = A008966(A037213(n)), when assumed A008966(0) = 0. - _Reinhard Zumkeller_, Jul 07 2013
%F A227291 a(n) = A063524(sum(A225817(n,k)*A225817(n,A000005(n)+1-k): k=1..A000005(n))). - _Reinhard Zumkeller_, Aug 01 2013
%F A227291 Multiplicative with a(p^e) = 1 if e=2, a(p^e) = 0 if e=1 or e>2. - _Antti Karttunen_, Jul 28 2017
%F A227291 Sum_{k=1..n} a(k) ~ 6*sqrt(n) / Pi^2. - _Vaclav Kotesovec_, Feb 02 2019
%F A227291 a(n) = A225569(A225546(n)-1). - _Peter Munn_, Oct 31 2019
%F A227291 From _Antti Karttunen_, Jul 18 2022: (Start)
%F A227291 a(n) = A010052(n) * A008966(A000196(n)).
%F A227291 a(n) = Sum_{d|n} A008836(n/d) * A307430(d).
%F A227291 a(n) = Sum_{d|n} A007427(n/d) * A322327(d).
%F A227291 (End)
%e A227291 a(3) = 0 because 3 is not the square of a squarefree number.
%e A227291 a(4) = 1 because sqrt(4) = 2, a squarefree number.
%p A227291 A227291 := proc(n)
%p A227291     local pe;
%p A227291     if n = 0 then
%p A227291         1;
%p A227291     else
%p A227291         for pe in ifactors(n)[2] do
%p A227291             if op(2,pe) <> 2 then
%p A227291                 return 0 ;
%p A227291             end if;
%p A227291         end do:
%p A227291     end if;
%p A227291     1 ;
%p A227291 end proc:
%p A227291 seq(A227291(n),n=1..100) ; # _R. J. Mathar_, Feb 07 2023
%t A227291 Table[Abs[Sum[MoebiusMu[n/d], {d,Select[Divisors[n], SquareFreeQ[#] &]}]], {n, 1, 200}] (* _Geoffrey Critzer_, Mar 18 2015 *)
%t A227291 Module[{nn=120,len,sfr},len=Ceiling[Sqrt[nn]];While[!SquareFreeQ[len],len++];sfr=(Select[Range[len],SquareFreeQ])^2; Table[If[MemberQ[ sfr,n],1,0],{n,nn}]] (* _Harvey P. Dale_, Nov 27 2024 *)
%o A227291 (PARI) a(n)=if(n<1, 0, direuler(p=2, n, 1+X^2)[n])
%o A227291 (PARI) A227291(n) = factorback(apply(e->(2==e), factor(n)[,2])); \\ _Antti Karttunen_, Jul 14 2022
%o A227291 (PARI) A227291(n) = (issquare(n) && issquarefree(sqrtint(n))); \\ _Antti Karttunen_, Jul 14 2022
%o A227291 (Haskell)
%o A227291 a227291 n = fromEnum $ (sum $ zipWith (*) mds (reverse mds)) == 1
%o A227291    where mds = a225817_row n
%o A227291 -- _Reinhard Zumkeller_, Jul 30 2013, Jul 07 2013
%o A227291 (Scheme) (define (A227291 n) (if (= 1 n) n (* (if (= 2 (A067029 n)) 1 0) (A227291 (A028234 n))))) ;; _Antti Karttunen_, Jul 28 2017
%Y A227291 Cf. A000196, A007427, A008836, A008683, A008966, A010052, A027750, A037213, A225546, A225569, A225817, A307430, A322327, A355448.
%Y A227291 Absolute values of A271102.
%K A227291 nonn,mult,easy
%O A227291 1,1
%A A227291 _Ralf Stephan_, Jul 05 2013