A289435 The arithmetic function v_3(n,3).

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

Offset: 2

Author

N. J. A. Sloane, Jul 06 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. A211316 (equals v_1(n,3)).

Programs

Maple

a:= n-> n*max(seq((floor((d-1-igcd(d, 3))/3)+1)
        /d, d=numtheory[divisors](n))):
seq(a(n), n=2..100);  # Alois P. Heinz, Jul 07 2017

Mathematica

a[n_]:=n*Max[Table[(Floor[(d - 1 - GCD[d, 3])/3] + 1)/d, {d, Divisors[n]}]]; Table[a[n], {n, 2, 100}] (* Indranil Ghosh, Jul 08 2017 *)

PARI

v(g,n,h)={my(t=0);fordiv(n,d,t=max(t,((d-1-gcd(d,g))\h + 1)*(n/d)));t}
a(n)=v(3,n,3); \\ Andrew Howroyd, Jul 07 2017

Python

from sympy import divisors, floor, gcd
def a(n): return n*max((floor((d - 1 - gcd(d, 3))/3) + 1)/d for d in divisors(n))
print([a(n) for n in range(2, 101)]) # Indranil Ghosh, Jul 08 2017

Extensions

a(41)-a(70) from Andrew Howroyd, Jul 07 2017