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 A115878 a(n) is the number of positive solutions of the Diophantine equation x^2 = y(y+n). 8
 0, 0, 1, 0, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 4, 2, 1, 2, 1, 1, 4, 1, 1, 4, 2, 1, 3, 1, 1, 4, 1, 3, 4, 1, 4, 2, 1, 1, 4, 4, 1, 4, 1, 1, 7, 1, 1, 7, 2, 2, 4, 1, 1, 3, 4, 4, 4, 1, 1, 4, 1, 1, 7, 4, 4, 4, 1, 1, 4, 4, 1, 7, 1, 1, 7, 1, 4, 4, 1, 7, 4, 1, 1, 4, 4, 1, 4, 4, 1, 7, 4, 1, 4, 1, 4, 10, 1, 2, 7, 2, 1, 4 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,9 COMMENTS Number of divisors d of n^2 such that d^2 < n^2 and n^2/d == d (mod 4). - Antti Karttunen, Oct 06 2018, based on Robert Israel's Jun 27 2014 comment in A115880. For odd n, a(n) can be computed from the prime signature. - David A. Corneth, Oct 07 2018 Number of r X s rectangles with integer side lengths such that r + s = n, r < s and (s-r) | (s*r). - Wesley Ivan Hurt, Apr 24 2020 LINKS Antti Karttunen, Table of n, a(n) for n = 1..18480 Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000 FORMULA From David A. Corneth, Oct 07 2018: (Start) a((2k+1) * 2^m) = floor(tau((2k + 1) ^ 2) / 2) for m <= 2. a((2k+1) * 2^m) = (2m - 3) * a(2k+1) + (m-2) for m > 2. (End) a(n) = Sum_{i=1..floor((n-1)/2)} chi(i*(n-i)/(n-2*i)), where chi is the integer characteristic. - Wesley Ivan Hurt, Apr 24 2020 EXAMPLE a(15) = 4 since there are 4 solutions (x,y) to x^2 = y(y+15), namely (4,1), (10,5), (18, 12) and (56, 49). Note how each x is obtained from each such divisor pair n2/d and d of n2 as (n2/d - d)/4, when their difference is a positive multiple of four, thus in case of n2 = 15^2 = 225 we get (225/1 - 1)/4 = 56, (225/3 - 3)/4 = 18, (225/5 - 5) = 10 and (225/9 - 9)/4 = 4. - Antti Karttunen, Oct 06 2018 a(96) = 10. We compute P, the largest power of 2 dividing n = 96. Then compute min(P, 4) and divide n by it. This gives 96/4 = 24. Then find the number of divisors of 24^2, which is 21. Dividing by 2 rounding down to the nearest integer gives 10, the value of a(96). - David A. Corneth, Oct 06 2018 MATHEMATICA a[n_] := Sum[Boole[d^2 < n^2 && Mod[n^2/d-d, 4] == 0], {d, Divisors[n^2]}]; Array[a, 102] (* Jean-François Alcover, Feb 27 2019, from PARI *) PROG (PARI) A115878(n) = { my(n2 = n^2); sumdiv(n2, d, ((d*d)>v)^2)>>1 \\ David A. Corneth, Oct 06 2018 CROSSREFS Cf. A067721, A115879, A115880, A115881. Sequence in context: A129192 A062540 A173636 * A127125 A256671 A327156 Adjacent sequences:  A115875 A115876 A115877 * A115879 A115880 A115881 KEYWORD nonn,easy AUTHOR Giovanni Resta, Feb 02 2006 STATUS approved

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Last modified May 26 13:49 EDT 2020. Contains 334626 sequences. (Running on oeis4.)