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

1,1

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

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)

Example

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

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 *)

Prog

(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
(Haskell)
a227291 n = fromEnum $ (sum $ zipWith (*) mds (reverse mds)) == 1
   where mds = a225817_row n
-- Reinhard Zumkeller, Jul 30 2013, Jul 07 2013
(Scheme) (define (A227291 n) (if (= 1 n) n (* (if (= 2 (A067029 n)) 1 0) (A227291 (A028234 n))))) ;; Antti Karttunen, Jul 28 2017

Crossrefs

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

Absolute values of A271102.

Sequence in context: A015179 A014504 A014999 * A271102 A353800 A353374

Adjacent sequences: A227288 A227289 A227290 * A227292 A227293 A227294

Keyword

nonn,mult,easy

Author

Ralf Stephan, Jul 05 2013

Status

approved