A368207 -id:A368207 - cp's OEIS frontend

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

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

A368582 a(n) = floor((sigma(n) + 1) / 2).

Original entry on oeis.org

1, 2, 2, 4, 3, 6, 4, 8, 7, 9, 6, 14, 7, 12, 12, 16, 9, 20, 10, 21, 16, 18, 12, 30, 16, 21, 20, 28, 15, 36, 16, 32, 24, 27, 24, 46, 19, 30, 28, 45, 21, 48, 22, 42, 39, 36, 24, 62, 29, 47, 36, 49, 27, 60, 36, 60, 40, 45, 30, 84, 31, 48, 52, 64, 42, 72, 34, 63

Offset: 1

Peter Luschny, Dec 31 2023

nonn

Cf. A000203, A000079 (2^n), A000396 (perfect), A088580, A317306, A368207 (Bacher).

using Nemo
A368582(n::Int) = div(divisor_sigma(n, 1) + 1, 2)
println([A368582(n) for n in 1:68])

Array[Floor[(DivisorSigma[1, #] + 1)/2] &, 120] (* Michael De Vlieger, Dec 31 2023 *)

a(n) = (sigma(n)+1)\2; \\ Michel Marcus, Jan 03 2024

a(p) = (p + 1) / 2 for all odd prime p.
a(n) = n <=> n term of union of A000079 and A000396. (If there are no odd perfect numbers also of A317306).
a(n) = floor(A088580(n)/2). - Omar E. Pol, Dec 31 2023

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Julia

Mathematica

Python