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

Author

Peter Luschny, Dec 26 2023

Keywords

• nonn

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.

Examples

			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.
		

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.

Crossrefs

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

Programs

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

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))

Formula

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