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%I #46 Feb 07 2023 09:42:26
%S 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 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 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 Characteristic function of squarefree numbers squared (A062503).
%H Antti Karttunen, <a href="/A227291/b227291.txt">Table of n, a(n) for n = 1..100000</a> (first 10000 terms from Reinhard Zumkeller)
%H <a href="/index/Ch#char_fns">Index entries for characteristic functions</a>
%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%F Dirichlet g.f.: zeta(2s)/zeta(4s) = prod[prime p: 1+p^(-2s) ], see A008966.
%F a(n) = A008966(A037213(n)), when assumed A008966(0) = 0. - _Reinhard Zumkeller_, Jul 07 2013
%F a(n) = A063524(sum(A225817(n,k)*A225817(n,A000005(n)+1-k): k=1..A000005(n))). - _Reinhard Zumkeller_, Aug 01 2013
%F 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 Sum_{k=1..n} a(k) ~ 6*sqrt(n) / Pi^2. - _Vaclav Kotesovec_, Feb 02 2019
%F a(n) = A225569(A225546(n)-1). - _Peter Munn_, Oct 31 2019
%F From _Antti Karttunen_, Jul 18 2022: (Start)
%F a(n) = A010052(n) * A008966(A000196(n)).
%F a(n) = Sum_{d|n} A008836(n/d) * A307430(d).
%F a(n) = Sum_{d|n} A007427(n/d) * A322327(d).
%F (End)
%e a(3) = 0 because 3 is not the square of a squarefree number.
%e a(4) = 1 because sqrt(4) = 2, a squarefree number.
%p A227291 := proc(n)
%p local pe;
%p if n = 0 then
%p 1;
%p else
%p for pe in ifactors(n)[2] do
%p if op(2,pe) <> 2 then
%p return 0 ;
%p end if;
%p end do:
%p end if;
%p 1 ;
%p end proc:
%p seq(A227291(n),n=1..100) ; # _R. J. Mathar_, Feb 07 2023
%t Table[Abs[Sum[MoebiusMu[n/d], {d,Select[Divisors[n], SquareFreeQ[#] &]}]], {n, 1, 200}] (* _Geoffrey Critzer_, Mar 18 2015 *)
%o (PARI) a(n)=if(n<1, 0, direuler(p=2, n, 1+X^2)[n])
%o (PARI) A227291(n) = factorback(apply(e->(2==e), factor(n)[,2])); \\ _Antti Karttunen_, Jul 14 2022
%o (PARI) A227291(n) = (issquare(n) && issquarefree(sqrtint(n))); \\ _Antti Karttunen_, Jul 14 2022
%o (Haskell)
%o a227291 n = fromEnum $ (sum $ zipWith (*) mds (reverse mds)) == 1
%o where mds = a225817_row n
%o -- _Reinhard Zumkeller_, Jul 30 2013, Jul 07 2013
%o (Scheme) (define (A227291 n) (if (= 1 n) n (* (if (= 2 (A067029 n)) 1 0) (A227291 (A028234 n))))) ;; _Antti Karttunen_, Jul 28 2017
%Y Cf. A000196, A007427, A008836, A008683, A008966, A010052, A027750, A037213, A225546, A225569, A225817, A307430, A322327, A355448.
%Y Absolute values of A271102.
%K nonn,mult,easy
%O 1,1
%A _Ralf Stephan_, Jul 05 2013
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