A368207 -id:A368207 - cp's OEIS frontend

A368457 a(n) = 2(Bacher(n) - sigma(n)) + n + 1 = 2(A368207(n) - A000203(n)) + n + 1.

2, 1, 0, 1, 0, -1, 0, -5, 2, -7, 0, -7, 0, -9, -4, -7, 0, -19, 0, -9, -10, -13, 0, -27, 4, -15, -12, -23, 0, -25, 0, -29, -14, -19, 0, -43, 0, -21, -16, -41, 0, -33, 0, -39, -28, -25, 0, -59, 6, -41, -20, -45, 0, -53, -16, -39, -22, -31, 0, -99, 0, -33, -20

Offset: 1

Peter Luschny, Dec 26 2023

sign

We can use the Bacher numbers A368207 to measure the primeness of a positive integer, similar to how the number of prime factors of an integer does, but based on the number of representations of n as w*x + y*z where max(w, x) < min(y, z).
Bacher's theorem shows that a(n) = 0 if n is an odd prime. Conversely, if a(n) = 0, we cannot conclude that n is prime as the example n = 35 shows, but this is probably the only exception.
Of the first 32,000 terms, approximately 88% are less than 0, 11% are equal to 0, and 1% are greater than 0. A368458 gives the indices for which a(n) is positive, and A368459 those for which a(n) is negative.
It appears that a(p^2) = p - 1 (A006093) for all prime p, following the observation by Knuth that apparently A368207(p^2) = (p^2 + 3*p)/2.

Peter Luschny, 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. A000203, A001358, A001248, A006093, A368207, A368458, A368459, A368460.

using Nemo
A368457(n) = 2 * (A368207(n) - divisor_sigma(n, 1)) + n + 1
println([A368457(n) for n in 1:63])

t[n_]:=t[n]=Select[Divisors[n], #^2<=n&];
A368207[n_]:=Sum[(1+Boole[d^2A368457[n_]:=2(A368207[n]-DivisorSigma[1,n])+n+1;
Array[A368457, 100] (* Paolo Xausa, Jan 02 2024 *)

A368341 Fixed points of A368207.

Original entry on oeis.org

0, 1, 2, 8, 9, 32, 128, 238, 512, 1012, 2048, 8192, 15070, 21658, 32768, 131072, 383548, 391612, 524288

Offset: 1

Chai Wah Wu, Dec 21 2023

Numbers k such that A368207(k)=k.

Conjecture: 2^(2k+1) for k>=0 (A004171) are terms.

Cf. A004171, A368207.

from itertools import count, islice
from sympy import divisors
def A368341_gen(startvalue=0): # generator of terms >= startvalue
    for n in count(max(startvalue,0)):
        c = 0
        for d2 in divisors(n):
            if d2**2 > n:
                break
            c += (d2<<2)-2 if d2**2n:
                break
        if c<=n:
            for wx in range(1,(n>>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
                            if c>n:
                                break
        if c==n:
            yield n
A368341_list = list(islice(A368341_gen(),10))

a(17)-a(19) from Chai Wah Wu, Dec 22 2023

A368496 Squarefree numbers k such that A368207(k^2) <> k+(sigma(k^2)-1)/2.

Original entry on oeis.org

30, 35, 42, 70, 105, 110, 130, 143, 154, 165, 182, 195, 210, 221, 238, 255, 266, 285, 286, 323, 330, 357, 374, 385, 390, 391, 399, 418, 429, 437, 442, 455, 462, 483, 494, 506, 510, 546, 570, 595, 598, 609, 638, 646, 663, 665, 667, 682, 690, 713, 714, 715, 754

Offset: 1

Chai Wah Wu, Dec 27 2023

nonn

Sequence suggested by the observation that A368207(k^2) = k+(sigma(k^2)-1)/2 for many numbers k.

Chai Wah Wu, Table of n, a(n) for n = 1..1071

Cf. A005117, A065764, A368207.

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

Original entry on oeis.org

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

A368458 Numbers k such that 2*(Bacher(k) - sigma(k)) + k + 1 > 0.

Original entry on oeis.org

1, 2, 4, 9, 25, 49, 121, 143, 169, 221, 289, 323, 361, 391, 437, 529, 667, 713, 841, 899, 961, 1073, 1147, 1271, 1333, 1369, 1517, 1591, 1681, 1739, 1763, 1849, 1927, 2021, 2173, 2209, 2279, 2491, 2537, 2773, 2809, 2867, 3127, 3233, 3481, 3551, 3599, 3721, 3763

Offset: 1

Peter Luschny, Dec 26 2023

nonn

We can use the Bacher numbers A368207 to measure the primeness of a positive integer, similar to how the number of prime factors of an integer does, but based on the number of representations of n as w*x + y*z with max(w, x) < min(y, z). Taking b(n) = 2*(Bacher(n) - sigma(n)) + n + 1 as the measure, Bacher's theorem shows that b(n) = 0 if n is an odd prime. Conversely, if b(n) = 0, we cannot conclude that n is prime as the example n = 35 shows, but this is probably the only exception.
This sequence gives the integers k for which b(k) is positive, and A368459 provides those for which b(k) is negative.
Apart from the first three terms, all terms are apparently odd semiprimes (A046315); they are odd composite numbers that cannot be divided by smaller composite numbers.

			To zoom into the internal order of the terms, the sequence can also be written as an irregular triangle (for n >= 3). It starts:
      4;
      9;
     25;
     49;
    121,  143;
    169,  221;
    289,  323;
    361,  391, 437;
    529,  667, 713;
    841,  899;
    961, 1073, 1147, 1271, 1333;
   1369, 1517, 1591;
   1681, 1739, 1763,
   1849, 1927, 2021, 2173;
A row contains the terms between consecutive squares of primes, p^2 included and p'^2 excluded. The first column is the squares of primes A001248. The length of the rows is A368460.

Peter Luschny, Table of n, a(n) for n = 1..1000
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. A000203, A046315, A001248, A368207, A368457, A368459, A368460.

using Nemo
function A368458List(slicenum::Int)
    results = [Int[] for _ in 1:slicenum + 1]
    slicelen = 1000
    Threads.@threads for sl in 1:slicenum
        first = (sl - 1) * slicelen + 1
        last = first + slicelen - 1
        result = results[sl]
        for n in first:2:last
            rem(n, 5) == 0 && continue
            if 2 * (divisor_sigma(n, 1) - A368207(n)) < n + 1
                push!(result, n)
    end end end
    results[slicenum + 1] = [2, 4, 25]
    sort(reduce(vcat, results))
end
print(A368458List(5)) # returns values up to param * 1000

from itertools import islice
def A368207(n):
    def t(n): return (d for d in divisors(n) if d*d <= n)
    def s(d): return 2*d - 1 if d*d == n else 4*d - 2
    def c(y, w, wx): return max(1, 2*((w*w < wx) + (y*y < n - wx)))
    return sum((sum(sum((c(y, w, wx) for y in t(n-wx) if wx < y*w), start=0)
    for w in t(wx)) for wx in range(1, n//2)),
    start=sum(s(d) for d in t(n)))
def A368457(n): return 2 * (A368207(n) - sigma(n)) + n + 1
def isA368458(n): return 0 < A368457(n)
def A368458Gen(n):
    while True:
        if isA368458(n): yield n
        n += 1
def A368458List(start, size): return list(islice(A368458Gen(start), size))
print(A368458List(1, 20))

k is a term <=> A368457(k) > 0 <=> 2*(A368207(k) - A000203(k)) + k + 1 > 0.

A368459 Numbers k such that 2*(Bacher(k) - sigma(k)) + k + 1 < 0.

Original entry on oeis.org

6, 8, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 33, 34, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95

Offset: 1

Peter Luschny, Dec 26 2023

nonn

Complementary to A368458, this sequence lists the indices of negative values of A368457. See the comments in A368458.

In summary, A368458 U A368459 U Primes U {35, ...} decomposes the positive integers into disjoint sets, whereby the nature of the fourth set is currently unclear; probably, it has only 35 as a member.

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. A000203, A100484, A001248, A368207, A368457, A368458, A368460.

println([n for n in 1:95 if A368457(n) < 0])

k is a term <=> A368457(k) < 0 <=> 2*(A368207(k) - A000203(k)) + k + 1 < 0.

A368460 a(n) = card(k: prime(n)^2 <= k < prime(n + 1)^2 and k term of A368458).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 3, 3, 2, 5, 3, 3, 4, 5, 4, 3, 9, 5, 4, 7, 5, 6, 14, 6, 4, 7, 3, 5, 22, 6, 6, 4, 16, 5, 12, 12, 8, 15, 13, 5, 19, 5, 10, 4, 30, 26, 10, 6, 12, 11, 5, 28, 19, 15, 20, 8, 19, 9, 7, 22

Offset: 1

Peter Luschny, Dec 26 2023

If A368458 is written as an irregular triangle for n >= 3, then a(n) is the length of row n.

Conjecture: For all n >= 5, there is at least one j such that b(j) = 2 * (Bacher(j) - sigma(j)) + j + 1 is > 0 and prime(n)^2 < b(j) < prime(n + 1)^2. In other words, a(n) > 1 for n >= 5.

			a(11) = 5 because 31^2 = 961, 1073, 1147, 1271, 1333, 1369 = 37^2 and all the terms are in that order in A368458.

Cf. A000203, A001248, A050216 (Brocard's Conjecture), A368207 (Bacher), A368457, A368458.

# using A368207
def A368460(n):
    pn = nth_prime(n); pn1 = nth_prime(n + 1)
    A368457 = lambda n: 2 * (A368207(n) - sigma(n)) + n + 1
    return sum(1 for n in range(pn ** 2, pn1 ** 2) if A368457(n) > 0)
print([A368460(n) for n in range(1, 25)])

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.

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

Original entry on oeis.org

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

cp's OEIS Frontend

A368457 a(n) = 2*(Bacher(n) - sigma(n)) + n + 1 = 2*(A368207(n) - A000203(n)) + n + 1.

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A368341 Fixed points of A368207.

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A368496 Squarefree numbers k such that A368207(k^2) <> k+(sigma(k^2)-1)/2.

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

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A368458 Numbers k such that 2*(Bacher(k) - sigma(k)) + k + 1 > 0.

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A368459 Numbers k such that 2*(Bacher(k) - sigma(k)) + k + 1 < 0.

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A368460 a(n) = card(k: prime(n)^2 <= k < prime(n + 1)^2 and k term of A368458).

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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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A368457 a(n) = 2(Bacher(n) - sigma(n)) + n + 1 = 2(A368207(n) - A000203(n)) + n + 1.

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