A007457 Number of (j,k): j+k=n, (j,n)=(k,n)=1, j,k squarefree. (Formerly M0224)

0, 1, 2, 2, 2, 2, 4, 4, 2, 2, 4, 4, 6, 4, 4, 6, 8, 6, 6, 6, 4, 8, 8, 8, 8, 8, 8, 6, 10, 8, 10, 10, 8, 12, 8, 10, 14, 12, 10, 12, 16, 10, 18, 14, 12, 14, 16, 14, 16, 14, 10, 16, 20, 14, 12, 16, 14, 20, 18, 14, 22, 20, 16, 20

Offset

1,3

Comments

Terms are even or 1: range = A004275. [Reinhard Zumkeller, Sep 26 2011]

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Ernesto Bruno Cossi, Joachim Herzog, Paul R. Smith and Richard Stong, Problem 6623, Amer. Math. Monthly, 99 (1992), 573-575.

R. G. Wilson v, Letter to N. J. A. Sloane, Oct. 1993

Formula

a(n) = Sum_{i=1..n-1} mu(i*(n-i))^2. - Ridouane Oudra, Nov 18 2019

Maple

with(numtheory): seq(add(mobius(i*(n-i))^2, i=1..n-1), n=1..80); # Ridouane Oudra, Nov 18 2019

Mathematica

a[n_] := Count[ Table[ If[ SquareFreeQ[j] && GCD[j, n] == 1, If[k = n-j; SquareFreeQ[k] && GCD[k, n] == 1, 1]], {j, 1, n-1}], 1]; Table[a[n], {n, 1, 64}](* Jean-François Alcover, Nov 28 2011 *)

Prog

(Haskell)
a007457 n = length [k | k <- [1..n-1], gcd k n == 1, a008966 k == 1,
                        let j = n - k, gcd j n == 1, a008966 j == 1]
-- Reinhard Zumkeller, Sep 26 2011
(Magma) f:=func<i, n|Gcd(i, n) eq 1 and IsSquarefree(i)>; [0] cat [#[i:i in [1..n-1]| f(i, n) and f(n-i, n) ]:n in [2..70]]; // Marius A. Burtea, Nov 19 2019
(Magma) [0] cat [&+[MoebiusMu(i*(n-i))^2:i in [1..n-1]]:n in [2..70]]; // Marius A. Burtea, Nov 19 2019

Crossrefs

Cf. A004275, A073311.

Sequence in context: A151565 A060632 A160407 * A119802 A237120 A060369

Adjacent sequences: A007454 A007455 A007456 * A007458 A007459 A007460

Keyword

nonn,nice,easy

Author

N. J. A. Sloane, Robert G. Wilson v

Status

approved