A370759 Numbers expressible in the form k*m + 2*(k+m) - 1, for positive k and m.

4, 7, 10, 11, 13, 15, 16, 19, 20, 22, 23, 25, 27, 28, 30, 31, 34, 35, 37, 39, 40, 43, 44, 45, 46, 47, 49, 50, 51, 52, 55, 58, 59, 60, 61, 63, 64, 65, 67, 70, 71, 72, 73, 75, 76, 79, 80, 82, 83, 85, 86, 87, 88, 90, 91, 93, 94, 95, 97, 99, 100, 103, 105, 106, 107, 109, 110, 111, 112

Offset

1,1

Comments

All such numbers are answers to the question: How many plane regions result from partitioning by two sets of straight lines, such that:

Each of the k straight lines of the first bundle passes though a single point A, and intersects each of the m straight lines of the second bundle each of which passes through a different point B. There are no straight lines belonging to both bundles, i.e. the line AB is not involved.

Because k*m+2*(k+m)-1 = (k+2)*(m+2)-5, and k and m are both positive, a(n) = A264828(n+2) - 5. - Kevin Ryde, Mar 26 2024

Links

Table of n, a(n) for n=1..69.

Nicolay Avilov, Explanatory drawing.

Nicolay Avilov, Illustration for terms a(1) - a(6).

Formula

If there are k straight lines in the first bundle and m straight lines in the second bundle, then we get k*m + 2*(k + m) - 1 regions.

Example

4 is a term: if each bundle consists of one straight line, the plane is divided into 4 regions.
7 is a term: if the first bundle consists of one line and the second consists of two lines, the plane is divided into 7 regions.
These and other examples are illustrated in the linked figures.

Prog

(PARI) print(Vec(setbinop((k, m)->k*m + 2*(k + m) - 1, [1..112]), 69)) \\ Michel Marcus, Mar 02 2024
(Python)
maxval = 112
av = [[k*m+2*k+2*m-1 for k in range(1, maxval)] for m in range(1, maxval)]
flat = [n for row in av for n in row]
uniq = list(set(flat))
a370759 = list(filter(lambda x: x<=maxval, uniq))
print(a370759)
# Robert Munafo, Mar 25 2024
(Python)
from itertools import count, islice
from sympy import isprime
def A370759_gen(startvalue=4): # generator of terms >= startvalue
    return filter(lambda n:not (isprime(n+5) or (n&1 and isprime((n>>1)+3))), count(max(startvalue, 4)))
A370759_list = list(islice(A370759_gen(), 20)) # Chai Wah Wu, Mar 26 2024

Crossrefs

Cf. A264828, A028875 (case when k=m).

Sequence in context: A123869 A309688 A072125 * A223024 A275340 A082206

Adjacent sequences: A370756 A370757 A370758 * A370760 A370761 A370762

Keyword

nonn

Author

Nicolay Avilov, Mar 01 2024

Status

approved