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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

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)

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-Fran├žois Alcover, Feb 27 2019, from PARI *)

PROG

(PARI) A115878(n) = { my(n2 = n^2); sumdiv(n2, d, ((d*d)<n2)&&(0==(((n2/d)-d)%4))); }; \\ Antti Karttunen, Oct 06 2018

(PARI) a(n) = my(v=min(2, valuation(n, 2))); numdiv((n>>v)^2)>>1 \\ David A. Corneth, Oct 06 2018

CROSSREFS

Cf. A067721, A115879, A115880, A115881.

Sequence in context: A129192 A062540 A173636 * A127125 A256671 A114171

Adjacent sequences:  A115875 A115876 A115877 * A115879 A115880 A115881

KEYWORD

nonn

AUTHOR

Giovanni Resta, Feb 02 2006

STATUS

approved

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Last modified May 21 15:00 EDT 2019. Contains 323443 sequences. (Running on oeis4.)