A377731 -id:A377731 - cp's OEIS frontend

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

Michel Marcus, Oct 25 2012

nonn

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

			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.

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.

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

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

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 *)

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

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

A377732 Numbers k such that max{d|k, d <= sqrt(k)} + min{d|k, d >= sqrt(k)} is a square.

Original entry on oeis.org

3, 4, 14, 18, 20, 39, 46, 55, 60, 63, 64, 94, 114, 136, 150, 154, 155, 156, 158, 183, 203, 243, 258, 275, 291, 295, 299, 308, 315, 320, 323, 324, 328, 334, 444, 446, 490, 544, 558, 570, 579, 580, 583, 584, 588, 594, 598, 600, 695, 710, 718, 799, 855, 878, 903, 904, 938, 943, 955, 959, 975, 978, 979, 988, 999

Offset: 1

Amiram Eldar, Nov 05 2024

nonn

Numbers k such that A063655(k) = A033676(k) + A033677(k) is a square.

The square terms of this sequence are the positive numbers of the form A141046(m) = 4*m^4.

Amiram Eldar, Table of n, a(n) for n = 1..10000
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.

Cf. A033676, A033677, A063655, A377731.

Subsequences: A141046 \ {0}, A377733, A377736.

q[k_] := If[IntegerQ[Sqrt[k]], IntegerQ[Sqrt[2*Sqrt[k]]], Module[{d = Divisors[k], nh}, nh = Length[d]/2; IntegerQ[Sqrt[d[[nh]] + d[[nh + 1]]]]]]; Select[Range[1000], q]

is(k) = if(issquare(k), issquare(2 * sqrtint(k)), my(d = divisors(k), nh = #d/2); issquare(d[nh] + d[nh + 1]));

from itertools import count, islice
from sympy import divisors
from sympy.ntheory.primetest import is_square
def A377732_gen(startvalue=1): # generator of terms >= startvalue
    for k in count(max(startvalue,1)):
        d = (a:=divisors(k))[len(a)-1>>1]
        if is_square(d+k//d):
            yield k
A377732_list = list(islice(A377732_gen(),30)) # Chai Wah Wu, Nov 06 2024

c * x^(3/4) / log(x) < R(x) < 2 * c * x^(3/4) / log(x) for sufficiently large x, where R(x) is the number of terms that do not exceed x, and c = A377731 (De Koninck et al., 2024).

A384563 Decimal expansion of Beta(1/4,1/4).

Original entry on oeis.org

7, 4, 1, 6, 2, 9, 8, 7, 0, 9, 2, 0, 5, 4, 8, 7, 6, 7, 3, 7, 3, 5, 4, 0, 1, 3, 8, 8, 7, 8, 1, 0, 4, 0, 1, 8, 4, 8, 7, 0, 3, 9, 5, 2, 9, 4, 0, 8, 7, 0, 6, 7, 6, 2, 2, 3, 4, 3, 7, 1, 2, 1, 8, 0, 2, 2, 4, 0, 8, 7, 1, 0, 7, 3, 5, 2, 4, 7, 9, 9, 1, 3, 4, 2, 9, 0, 8, 7, 4, 4, 6, 6, 0, 1, 4, 8, 7, 5, 8, 9

Offset: 1

Stefano Spezia, Jun 03 2025

			7.416298709205487673735401388781040184870395294...

Michael I. Shamos, A catalog of the real numbers, (2007). See p. 406.

Similar constants Beta(1/k,1/k): A000796 (k=2), A197374 (k=3).

Cf. A002161, A068466, A175576, A377731.

Mathematica
```
RealDigits[Beta[1/4,1/4],10,100][[1]]
```

Equals Gamma(1/4)^2/sqrt(Pi) = A068466^2/A002161.

Equals 2*A175576 = 3*A377731. - Hugo Pfoertner, Jun 03 2025

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

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A377732 Numbers k such that max{d|k, d <= sqrt(k)} + min{d|k, d >= sqrt(k)} is a square.

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Mathematica

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A384563 Decimal expansion of Beta(1/4,1/4).

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Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula