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

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

1,2

Comments

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.

Links

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.

Formula

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

Example

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.

Prog

(SageMath)
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))
(Julia)
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

Crossrefs

Cf. A000203, A046315, A001248, A368207, A368457, A368459, A368460.

Sequence in context: A013168 A013183 A049963 * A114110 A337693 A335342

Adjacent sequences: A368455 A368456 A368457 * A368459 A368460 A368461

Keyword

nonn

Author

Peter Luschny, Dec 26 2023

Status

approved