A368276 -id:A368276 - cp's OEIS frontend

A368277 Prime numbers that have an even number of monotone Bacher representations (A368276).

5, 7, 13, 17, 23, 43, 53, 59, 61, 71, 79, 83, 107, 109, 113, 127, 131, 137, 139, 167, 181, 191, 193, 199, 211, 223, 227, 239, 241, 257, 271, 277, 293, 307, 313, 317, 331, 337, 347, 353, 359, 367, 379, 389, 401, 421, 431, 439, 449, 457, 461, 467, 479, 499

Offset: 1

Peter Luschny, Dec 19 2023

nonn

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], 2022.

Cf. A368276, A368278, A368207.

using Nemo
println([n for n in 1:500 if iseven(A368276(n)) && is_prime(n)])

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

from itertools import takewhile, islice
from sympy import nextprime, divisors
def A368277_gen(startvalue=2): # generator of terms >= startvalue
    p = max(nextprime(startvalue-1),2)
    while True:
        c = sum(takewhile(lambda x:x**2<=p,divisors(p))) &1
        for wx in range(1,(p>>1)+1):
            for d1 in divisors(wx):
                if d1**2 > wx:
                    break
                m = p-wx
                c = c+sum(1 for d in takewhile(lambda x:x**2<=m,divisors(m)) if wxA368277_list = list(islice(A368277_gen(),30)) # Chai Wah Wu, Dec 19 2023

A368278 Prime numbers that have an odd number of monotone Bacher representations (A368276).

Original entry on oeis.org

2, 3, 11, 19, 29, 31, 37, 41, 47, 67, 73, 89, 97, 101, 103, 149, 151, 157, 163, 173, 179, 197, 229, 233, 251, 263, 269, 281, 283, 311, 349, 373, 383, 397, 409, 419, 433, 443, 463, 487, 491, 521, 523, 557, 577, 587, 601, 607, 619, 659, 661, 673, 677, 701, 719

Offset: 1

Peter Luschny, Dec 19 2023

nonn

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 = 19, the 5 solutions are (w, x, y, z) = (0, 0, 1, 19), (1, 1, 2, 9), (1, 1, 3, 6), (1, 3, 4, 4), (2, 2, 3, 5).

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. A368276, A368277, A368207.

using Nemo
println([n for n in 1:720 if isodd(A368276(n)) && is_prime(n)])

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

from itertools import takewhile, islice
from sympy import divisors, nextprime
def A368278_gen(startvalue=2): # generator of terms >= startvalue
    p = max(nextprime(startvalue-1),2)
    while True:
        c = sum(takewhile(lambda x:x**2<=p,divisors(p))) & 1
        for wx in range(1,(p>>1)+1):
            for d1 in divisors(wx):
                if d1**2 > wx:
                    break
                m = p-wx
                c = c+sum(1 for d in takewhile(lambda x:x**2<=m,divisors(m)) if wxA368278_list = list(islice(A368278_gen(),30)) # Chai Wah Wu, Dec 19 2023

A368207 Bacher numbers: number of nonnegative representations of n = wx+yz with max(w,x) < min(y,z).

Original entry on oeis.org

1, 2, 2, 5, 3, 8, 4, 8, 9, 9, 6, 18, 7, 12, 14, 19, 9, 20, 10, 27, 16, 18, 12, 34, 20, 21, 20, 30, 15, 44, 16, 32, 24, 27, 30, 51, 19, 30, 28, 49, 21, 58, 22, 42, 41, 36, 24, 70, 35, 47, 36, 49, 27, 66, 36, 72, 40, 45, 30, 88, 31, 48, 62, 71, 42, 74, 34, 63, 48

Offset: 1

Don Knuth, Dec 16 2023

nonn

When n = p is an odd prime, Bacher proved that a(p) = (p+1)/2.
It appears that also a(k*p) = sigma(k)*(p+1)/2, for all prime p > 2k, where sigma(k) is the sum of the divisors of k (A000203).
It appears furthermore that a(p^2) = (p^2 + 3*p)/2; a(p^3) = (p^3 + p^2 + p + 1)/2; a(p^4) = (p^4 + p^3 + 3p^2 + p)/2, for all prime p.
Conjectures: (1) a(n) >= sigma(n)/2, with equality if and only if n has no middle divisors, i.e., if and only if n is in A071561. (2) a(n)/sigma(n) converges to 1/2. - Pontus von Brömssen, Dec 18 2023
From Chai Wah Wu, Dec 19 2023: (Start)
Considering representations where min(w,x)=0 shows that a(n) >= 2*A066839(n) - A038548(n).
It appears that a(p^5) = (p^5 + p^4 + p^3 + p^2 + p + 1)/2 for all prime p > 2 and a(p^6) = (p^6 + p^5 + p^4 + 3p^3 + p^2 + p)/2 for all prime p.
Conjecture: a(p^m) = sigma(p^m)/2 for odd m and all prime p > 2. a(p^m) = (sigma(p^m)-1)/2 + p^(m/2) for even m and all prime p. a(2^m) = sigma(2^m)/2 + 1/2 for m odd. (End)

			For n = 13, the a(13) = 7 solutions are (w,x,y,z) = (0,0,1,13), (0,0,13,1), (1,1,2,6), (1,1,3,4), (1,1,4,3), (1,1,6,2), (2,2,3,3).

Don Knuth, 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.
Michael S. Branicky, Python translation of Knuth's CWEB program
Pontus von Brömssen, Plot of (n, a(n) - sigma(n)/2) for n = 1..100000.
Pontus von Brömssen, Plot of (n, b(n)) for n = 1..100000, where a(n) = sigma(n)/2 + b(n)*sqrt(n).
Don Knuth, CWEB program.
Peter Luschny, A Formula for the Bacher Numbers.

Cf. A000203, A071561, A368276 (monotone), A368341 (fixed points), A368457, A368458 (semiprimes), A368580 (degenerated).

CWEB
```
@ See Knuth link.
```

function A368207(n)
    t(n) = (d for d in divisors(n) if d * d <= n)
    s(d) = d * d == n ? d * 2 - 1 : d * 4 - 2
    c(y, w, wx) = max(1, 2*(Int(w*w < wx) + Int(y*y < n - wx)))
    sum(sum(sum(c(y, w, wx) for y in t(n - wx) if wx < y * w; init=0)
    for w in t(wx)) for wx in 1:div(n, 2); init=sum(s(d) for d in t(n)))
end
println([A368207(n) for n in 1:69])  # Peter Luschny, Dec 21 2023

t[n_] := t[n] = Select[Divisors[n], #^2 <= n&];
A368207[n_] := Sum[(1 + Boole[d^2 < n])(2d - 1),{d, t[n]}] + Sum[If[wx < y*w, Max[1, 2(Boole[w^2 < wx] + Boole[y^2 < n-wx])], 0], {wx, Floor[n/2]},{w, t[wx]}, {y, t[n - wx]}];
Array[A368207, 100] (* Paolo Xausa, Jan 02 2024 *)

# See Branicky link for translation of Knuth's CWEB program.

from math import isqrt
def A368207(n):
    c, r = 0, isqrt(n)
    for w in range(r+1):
        for x in range(w,r+1):
            wx = w*x
            if wx>n:
                break
            for y in range(x+1,r+1):
                for z in range(y,n+1):
                    yz = wx+y*z
                    if yz>n:
                        break
                    if yz==n:
                        m = 1
                        if w!=x:
                            m<<=1
                        if y!=z:
                            m<<=1
                        c+=m
    return c # Chai Wah Wu, Dec 19 2023

from sympy import divisors
# faster program
def A368207(n):
    c = 0
    for d2 in divisors(n):
        if d2**2 > n:
            break
        c += (d2<<2)-2 if d2**2>1)+1):
        for d1 in divisors(wx):
            if d1**2 > wx:
                break
            for d2 in divisors(m:=n-wx):
                if d2**2 > m:
                    break
                if wx < d1*d2:
                    k = 1
                    if d1**2 != wx:
                        k <<=1
                    if d2**2 != m:
                        k <<=1
                    c+=k
    return c # Chai Wah Wu, Dec 19 2023

Let t(n) = {d|n and d*d <= n}, and s(d, n) = 2*d - 1 if d*d = n, otherwise 4*d - 2. Then a(n) = (Sum_{d in t(n)} s(d, n)) + (Sum_{k=1..floor(n/2)} Sum_{w in t(k)} Sum_{y in t(n-k) and k < y*w} max(1, 2*([w*w < k] + [y*y < n - k]))), where [] denote the Iverson brackets. (See the 'Julia' implementation below.) - Peter Luschny, Dec 21 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).

A368581 The sum of weights of nondegenerated monotone Bacher representations of n.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 2, 0, 2, 1, 4, 0, 5, 4, 2, 4, 7, 2, 8, 5, 4, 10, 10, 2, 9, 13, 8, 8, 13, 8, 14, 10, 12, 19, 10, 8, 17, 22, 16, 9, 19, 18, 20, 20, 11, 28, 22, 16, 20, 21, 24, 27, 25, 26, 16, 24, 28, 37, 28, 16, 29, 40, 24, 34, 22, 34, 32, 41, 36, 28, 34, 28

Offset: 1

Peter Luschny, Dec 31 2023

nonn

For the definition of 'Bacher representation' and related notions, see the comments in A368580.

			See the example in A368580.

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, A368580.

using Nemo
function A368581(n::Int)
    t(n) = (d for d in divisors(n) if d * d <= n)
    c(y, w, wx) = max(1, 2 * (Int(w * w < wx) + Int(y * y < n - wx)))
    sum(sum(sum(c(y, w, wx) for y in t(n - wx) if wx < y * w; init=0)
    for w in t(wx)) for wx in 1:div(n, 2); init=0)
end
println([A368581(n) for n in 1:72])

t[n_]:=t[n]=Select[Divisors[n],#^2<=n&];
A368581[n_]:=Sum[If[wxA368581,100] (* Paolo Xausa, Jan 02 2024 *)

a(n) = Sum_{k in K} Sum_{w in W} Sum_{y in Y} max(1, 2*([w^2 < k] + [y^2 < n - k])), where the square brackets denote Iverson brackets and k in K <=> 1 <= k <= floor(n/2), w in W <=> w|k and w^2 <= k, and y in Y <=> y|n-k and y^2 <= n-k and k < y*w. (See the Julia implementation.)
a(n) + A368580(n) = A368207(n).
a(p) = (p + 1) / 2 - 2 for all odd prime p.

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A368277 Prime numbers that have an even number of monotone Bacher representations (A368276).

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A368278 Prime numbers that have an odd number of monotone Bacher representations (A368276).

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A368207 Bacher numbers: number of nonnegative representations of n = w*x+y*z with max(w,x) < min(y,z).

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CWEB

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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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A368581 The sum of weights of nondegenerated monotone Bacher representations of n.

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A368207 Bacher numbers: number of nonnegative representations of n = wx+yz with max(w,x) < min(y,z).