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

1,2

Comments

For more information about the quarter-squares see A002620.

Links

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

Wikipedia, Table of divisors.

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

Example

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.

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 *)

Prog

(Haskell)
a239930 = sum . map a240025 . a027750_row
-- Reinhard Zumkeller, Jul 05 2014
(PARI) a(n) = sumdiv(n, d, issquare(d) + issquare(4*d + 1)); \\ Amiram Eldar, Dec 31 2023

Crossrefs

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

Cf. A240025.

Sequence in context: A029242 A029236 A152188 * A226859 A025820 A109704

Adjacent sequences: A239927 A239928 A239929 * A239931 A239932 A239933

Keyword

nonn

Author

Omar E. Pol, Jun 19 2014

Status

approved