A239930 Number of distinct quarter-squares dividing n.

1, 2, 1, 3, 1, 3, 1, 3, 2, 2, 1, 5, 1, 2, 1, 4, 1, 4, 1, 4, 1, 2, 1, 5, 2, 2, 2, 3, 1, 4, 1, 4, 1, 2, 1, 7, 1, 2, 1, 4, 1, 4, 1, 3, 2, 2, 1, 6, 2, 3, 1, 3, 1, 4, 1, 4, 1, 2, 1, 7, 1, 2, 2, 5, 1, 3, 1, 3, 1, 2, 1, 8, 1, 2, 2, 3, 1, 3, 1, 5, 3, 2, 1, 6, 1, 2, 1, 3, 1, 6, 1, 3, 1, 2, 1, 6, 1, 3, 2, 6, 1, 3, 1, 3, 1, 2, 1, 7, 1, 3

Offset: 1

Author

Omar E. Pol, Jun 19 2014

Keywords

• nonn

Comments

For more information about the quarter-squares see A002620.

Examples

			For n = 12 the quarter-squares <= 12 are [0, 0, 1, 2, 4, 6, 9, 12]. There are five quarter-squares that divide 12; they are [1, 2, 4, 6, 12], so a(12) = 5.
		

Links

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

Crossrefs

Cf. A000005, A001221, A001511, A002620, A005086, A006519, A007862, A013661, A027750, A046951, A147645, A236103.

Cf. A240025.

Programs

Haskell

a239930 = sum . map a240025 . a027750_row
-- Reinhard Zumkeller, Jul 05 2014

Maple

isA002620 := proc(n)
    local k,qsq ;
    for k from 0 do
        qsq := floor(k^2/4) ;
        if n = qsq then
            return true;
        elif qsq > n then
            return false;
        end if;
    end do:
end proc:
A239930 := proc(n)
    local a,d ;
    a :=0 ;
    for d in numtheory[divisors](n) do
        if isA002620(d) then
            a:= a+1 ;
        end if;
    end do:
    a;
end proc: # R. J. Mathar, Jul 03 2014

Mathematica

qsQ[n_] := AnyTrue[Range[Ceiling[2 Sqrt[n]]], n == Floor[#^2/4]&]; a[n_] := DivisorSum[n, Boole[qsQ[#]]&]; Array[a, 110] (* Jean-François Alcover, Feb 12 2018 *)

PARI

a(n) = sumdiv(n, d, issquare(d) + issquare(4*d + 1)); \\ Amiram Eldar, Dec 31 2023

Formula

a(n) = Sum_{k=1..A000005(n)} A240025(A027750(n,k)). - Reinhard Zumkeller, Jul 05 2014

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(2) + 1 = A013661 + 1 = 2.644934... . - Amiram Eldar, Dec 31 2023