A291304 The arithmetic function v_5(n,2).

1, 1, 2, 0, 3, 3, 4, 4, 5, 5, 6, 6, 7, 5, 8, 8, 9, 9, 10, 10, 11, 11, 12, 10, 13, 13, 14, 14, 15, 15, 16, 16, 17, 15, 18, 18, 19, 19, 20, 20, 21, 21, 22, 20, 23, 23, 24, 24, 25, 25, 26, 26, 27, 25, 28, 28, 29, 29, 30, 30, 31, 31, 32, 30, 33, 33, 34, 34, 35

Offset: 2

Author

Robert Price, Aug 21 2017

Keywords

• nonn

References

• J. Butterworth, Examining the arithmetic function v_g(n,h). Research Papers in Mathematics, B. Bajnok, ed., Gettysburg College, Vol. 8 (2008).

Links

• Bela Bajnok, Additive Combinatorics: A Menu of Research Problems, arXiv:1705.07444 [math.NT], May 2017. See Table in Section 1.6.1.

Crossrefs

Cf. A289435, A289436, A289437, A289438, A289439, A289440, A289441, A001622.

Programs

Mathematica

v[g_, n_, h_] := (d = Divisors[n]; Max[(Floor[(d - 1 - GCD[d, g])/h] + 1)*n/d]); Table[v[5, n, 2], {n, 2, 70}]

Formula

a(n) = (n + gcd(n,10) - 2*gcd(n,5))/2. - Ridouane Oudra, Feb 17 2025

Sum_{n>=6} (-1)^(n+1)/a(n) = 1/2 - sqrt(1-2/sqrt(5))*Pi/10 + sqrt(5)*log(phi)/10 - log(5)/4, where phi is the golden ratio (A001622). - Amiram Eldar, Feb 20 2025