A368581 -id:A368581 - cp's OEIS frontend

A368276 Number of nonnegative representations of n = wx + yz with max(w, x) < min(y, z) and w <= x <= y <= z.

1, 1, 1, 3, 2, 3, 2, 3, 5, 4, 3, 6, 4, 4, 5, 9, 4, 7, 5, 9, 6, 7, 4, 11, 11, 7, 7, 11, 7, 13, 7, 10, 9, 10, 9, 19, 9, 9, 10, 17, 9, 17, 8, 14, 14, 13, 7, 21, 17, 14, 13, 17, 10, 20, 13, 22, 14, 15, 10, 26, 14, 13, 18, 28, 15, 22, 13, 19, 17, 25, 12, 33, 15, 18

Offset: 1

Peter Luschny, Dec 19 2023

nonn

Number of monotone Bacher representations (A368207) of n. We call a quadruple (w, x, y, z) of nonnegative integers a monotone Bacher representation of n if and only if n = w*x + y*z and w <= x < y <= z.

			For n = 13, the 4 solutions are (w, x, y, z) = (0, 0, 1, 13), (1, 1, 2, 6), (1, 1, 3, 4), (2, 2, 3, 3).

Roland Bacher, A quixotic proof of Fermat's two squares theorem for prime numbers, American Mathematical Monthly, Vol. 130, No. 9 (November 2023), 824-836; arXiv version, arXiv:2210.07657 [math.NT], 2023.

Cf. A368207, A368276, A368457, A368458, A368459, A368580, A368581.

using Nemo
function A368276(n)
    t(n) = (d for d in divisors(n) if d * d <= n)
    sum(sum(sum(1 for d in t(n - wx) if wx < d * dx; init=0)
            for dx in t(wx)) for wx in 1:div(n, 2); init=sum(t(n)))
end
println([A368276(n) for n in 1:74])  # Peter Luschny, Dec 19 2023

t[n_]:=t[n]=Select[Divisors[n],#^2<=n&];
A368276[n_]:=Total[t[n]]+Sum[Boole[wxA368276,100] (* Paolo Xausa, Jan 02 2024 *)

from itertools import takewhile
from sympy import divisors
def A368276(n):
    c = sum(takewhile(lambda x: x**2 <= n, divisors(n)))
    for wx in range(1, (n >> 1) + 1):
        for d1 in divisors(wx):
            if d1**2 > wx:
                break
            m = n - wx
            c += sum(1
                for d in takewhile(lambda x: x**2 <= m, divisors(n - wx))
                if wx < d * d1)
    return c  # Chai Wah Wu, Dec 19 2023

A368580 a(n) = Sum_{d|n and d^2 <= n} (1 + [d^2 < n]) * (2*d - 1), where [.] denote the Iverson brackets.

Original entry on oeis.org

1, 2, 2, 5, 2, 8, 2, 8, 7, 8, 2, 18, 2, 8, 12, 15, 2, 18, 2, 22, 12, 8, 2, 32, 11, 8, 12, 22, 2, 36, 2, 22, 12, 8, 20, 43, 2, 8, 12, 40, 2, 40, 2, 22, 30, 8, 2, 54, 15, 26, 12, 22, 2, 40, 20, 48, 12, 8, 2, 72, 2, 8, 38, 37, 20, 40, 2, 22, 12, 52, 2, 84, 2, 8

Offset: 1

Peter Luschny, Dec 31 2023

nonn

A quadruple (w, x, y, z) of nonnegative integers is a 'Bacher representation' of n if and only if n = w*x + y*z and max(w,x) < min(y,z).
A Bacher representation is 'monotone' if additionally w <= x <= y <= z.
A Bacher representation is 'degenerated' if w = 0. The weight of a Bacher representation is defined as
  W(w, x, y, z) = max(1, 2*([w < x] + [y < z])).
a(n) is the sum of the weights of all degenerated monotone Bacher representations of n. The complementary sum of weights of nondegenerated monotone Bacher representations is A368581.

			Below are the monotone Bacher representations of n = 27 listed.
  W(0, 0, 1, 27) = 2;
  W(0, 0, 3,  9) = 2;
  W(0, 1, 3,  9) = 4;
  W(0, 2, 3,  9) = 4;
  W(1, 1, 2, 13) = 2;
  W(1, 2, 5,  5) = 2;
  W(1, 3, 4,  6) = 4.
Thus a(27) = 2 + 2 + 4 + 4 = 12. Adding all weights gives A368207(27) = 20.
For instance, the integers n = 6, 8, and 12 have only degenerated Bacher representation, so for these cases, a(n) = A368207(n).

Paolo Xausa, Table of n, a(n) for n = 1..10000
Roland Bacher, A quixotic proof of Fermat's two squares theorem for prime numbers, American Mathematical Monthly, Vol. 130, No. 9 (November 2023), 824-836; arXiv version, arXiv:2210.07657 [math.NT], 2022.

Cf. A368207, A368276, A368581.

using Nemo
function A368580(n)
    sum(d * d == n ? d * 2 - 1 : d * 4 - 2
    for d in (d for d in divisors(n) if d * d <= n))
end
println([A368580(n) for n in 1:74])

A368580[n_]:=DivisorSum[n,(1+Boole[#^2A368580,100] (* Paolo Xausa, Jan 01 2024 *)

a(p) = 2 for all prime p.
a(n) is odd if and only if n is a square.
a(n) + A368581(n) = A368207(n).

cp's OEIS Frontend

A368276 Number of nonnegative representations of n = wx + yz with max(w, x) < min(y, z) and w <= x <= y <= z.

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A368580 a(n) = Sum_{d|n and d^2 <= n} (1 + [d^2 < n]) * (2*d - 1), where [.] denote the Iverson brackets.

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cp's OEIS Frontend

A368276 Number of nonnegative representations of n = w*x + y*z with max(w, x) < min(y, z) and w <= x <= y <= z.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Julia

Mathematica

Python

A368580 a(n) = Sum_{d|n and d^2 <= n} (1 + [d^2 < n]) * (2*d - 1), where [.] denote the Iverson brackets.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Julia

Mathematica

Formula

A368276 Number of nonnegative representations of n = wx + yz with max(w, x) < min(y, z) and w <= x <= y <= z.