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Number of distinct quarter-squares dividing n.
4

%I #33 Dec 31 2023 06:23:53

%S 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,

%T 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,

%U 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

%N Number of distinct quarter-squares dividing n.

%C For more information about the quarter-squares see A002620.

%H Reinhard Zumkeller, <a href="/A239930/b239930.txt">Table of n, a(n) for n = 1..10000</a>

%H Wikipedia, <a href="http://www.wikipedia.org/wiki/Table_of_divisors">Table of divisors</a>.

%F a(n) = Sum_{k=1..A000005(n)} A240025(A027750(n,k)). - _Reinhard Zumkeller_, Jul 05 2014

%F 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

%e 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.

%p isA002620 := proc(n)

%p local k,qsq ;

%p for k from 0 do

%p qsq := floor(k^2/4) ;

%p if n = qsq then

%p return true;

%p elif qsq > n then

%p return false;

%p end if;

%p end do:

%p end proc:

%p A239930 := proc(n)

%p local a,d ;

%p a :=0 ;

%p for d in numtheory[divisors](n) do

%p if isA002620(d) then

%p a:= a+1 ;

%p end if;

%p end do:

%p a;

%p end proc: # _R. J. Mathar_, Jul 03 2014

%t 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 *)

%o (Haskell)

%o a239930 = sum . map a240025 . a027750_row

%o -- _Reinhard Zumkeller_, Jul 05 2014

%o (PARI) a(n) = sumdiv(n, d, issquare(d) + issquare(4*d + 1)); \\ _Amiram Eldar_, Dec 31 2023

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

%Y Cf. A240025.

%K nonn

%O 1,2

%A _Omar E. Pol_, Jun 19 2014