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

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

Author

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

Keywords

• nonn
• nice
• easy

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

Crossrefs

Cf. A004275, A073311.

Programs

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; [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

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 *)

Formula

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