A289437 The arithmetic function v_2(n,4).

0, 1, 1, 1, 2, 2, 2, 3, 2, 3, 4, 3, 4, 5, 4, 4, 6, 5, 5, 7, 6, 6, 8, 6, 6, 9, 8, 7, 10, 8, 8, 11, 8, 10, 12, 9, 10, 13, 10, 10, 14, 11, 12, 15, 12, 12, 16, 14, 12, 17, 13, 13, 18, 15, 16, 19, 14, 15, 20, 15, 16, 21, 16, 16, 22, 17, 17, 23, 20

Offset: 2

Author

N. J. A. Sloane, Jul 07 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. A289436, A289438.

Programs

Maple

a:= n-> n*max(seq((floor((d-1-igcd(d, 2))/4)+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, 2])/4] + 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(2,n,4); \\ Andrew Howroyd, Jul 07 2017

Python

from sympy import divisors, floor, gcd
def a(n): return n*max([(floor((d - 1 - gcd(d, 2))/4) + 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