A227291 Characteristic function of squarefree numbers squared (A062503).

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, 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, 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

Offset: 1

Author

Ralf Stephan, Jul 05 2013

Keywords

• nonn
• mult
• easy

Examples

			a(3) = 0 because 3 is not the square of a squarefree number.
a(4) = 1 because sqrt(4) = 2, a squarefree number.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..100000 (first 10000 terms from Reinhard Zumkeller)
• Index entries for characteristic functions
• Index entries for sequences computed from exponents in factorization of n

Crossrefs

Cf. A000196, A007427, A008836, A008683, A008966, A010052, A027750, A037213, A225546, A225569, A225817, A307430, A322327, A355448.

Absolute values of A271102.

Programs

Haskell

a227291 n = fromEnum $ (sum $ zipWith (*) mds (reverse mds)) == 1
   where mds = a225817_row n
-- Reinhard Zumkeller, Jul 30 2013, Jul 07 2013

Maple

A227291 := proc(n)
    local pe;
    if n = 0 then
        1;
    else
        for pe in ifactors(n)[2] do
            if op(2,pe) <> 2 then
                return 0 ;
            end if;
        end do:
    end if;
    1 ;
end proc:
seq(A227291(n),n=1..100) ; # R. J. Mathar, Feb 07 2023

Mathematica

Table[Abs[Sum[MoebiusMu[n/d], {d,Select[Divisors[n], SquareFreeQ[#] &]}]], {n, 1, 200}] (* Geoffrey Critzer, Mar 18 2015 *)
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 *)

PARI

a(n)=if(n<1, 0, direuler(p=2, n, 1+X^2)[n])

PARI

A227291(n) = factorback(apply(e->(2==e), factor(n)[,2])); \\ Antti Karttunen, Jul 14 2022

PARI

A227291(n) = (issquare(n) && issquarefree(sqrtint(n))); \\ Antti Karttunen, Jul 14 2022

Scheme

(define (A227291 n) (if (= 1 n) n (* (if (= 2 (A067029 n)) 1 0) (A227291 (A028234 n))))) ;; Antti Karttunen, Jul 28 2017

Formula

Dirichlet g.f.: zeta(2s)/zeta(4s) = prod[prime p: 1+p^(-2s) ], see A008966.

a(n) = A008966(A037213(n)), when assumed A008966(0) = 0. - Reinhard Zumkeller, Jul 07 2013

a(n) = A063524(sum(A225817(n,k)*A225817(n,A000005(n)+1-k): k=1..A000005(n))). - Reinhard Zumkeller, Aug 01 2013

Multiplicative with a(p^e) = 1 if e=2, a(p^e) = 0 if e=1 or e>2. - Antti Karttunen, Jul 28 2017

Sum_{k=1..n} a(k) ~ 6*sqrt(n) / Pi^2. - Vaclav Kotesovec, Feb 02 2019

a(n) = A225569(A225546(n)-1). - Peter Munn, Oct 31 2019

From Antti Karttunen, Jul 18 2022: (Start)

a(n) = A010052(n) * A008966(A000196(n)).

a(n) = Sum_{d|n} A008836(n/d) * A307430(d).

a(n) = Sum_{d|n} A007427(n/d) * A322327(d).

(End)