A211996 Number of ordered pairs (i,j) such that i*j=n and i+j is a square.

0, 0, 2, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 2, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 4, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 4, 1, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0

Offset: 1

Author

Michel Marcus, Oct 25 2012

Keywords

• nonn

Comments

a(n) = 1 for n > 0 in A141046.

a(8820) = 8 and it is the only term in the first 10000 terms that is greater than 6. There are 977 terms in the first 10000 terms that are greater than zero. - Harvey P. Dale, Nov 08 2012

Examples

			For n=3, the pairs (a,b) such that a*b=3 are (1,3) and (3,1). Both pairs add up to a square, so a(3) = 2.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• David Clark, An arithmetical function associated with the rank of elliptic curves, Canad. Math. Bull. Vol. 34 (2), (1991), pp. 181-185.
• Jean-Marie De Koninck, A. Arthur Bonkli Razafindrasoanaivolala, and Hans Schmidt Ramiliarimanana, Integers with a sum of co-divisors yielding a square, Research in Number Theory, Vol. 10, No. 2 (2024), Article 30; author's copy.

Crossrefs

Cf. A010052, A141046, A218381, A218382, A377731.

Programs

Haskell

a211996 n = length [x | x <- [1..n], let (y, m) = divMod n x,
                        m == 0, a010052 (x + y) == 1]
-- Reinhard Zumkeller, Oct 28 2012

Mathematica

nop[n_]:=Module[{divs=Divisors[n]},Count[Thread[{divs,Reverse[divs]}], ?(IntegerQ[Sqrt[Total[#]]]&)]]; Array[nop,90] (* _Harvey P. Dale, Nov 08 2012 *)
a[n_] := DivisorSum[n, 1 &, IntegerQ[Sqrt[# + n/#]] &]; Array[a, 100] (* Amiram Eldar, Nov 05 2024 *)

PARI

a(n) = sumdiv(n, d, issquare(d+n/d)); \\ Michel Marcus, Jan 18 2021

Formula

Sum_{k=1..n} a(k) = c * n^(3/4) + O(sqrt(n)), where c = A377731 (De Koninck et al., 2024). - Amiram Eldar, Nov 05 2024