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

Author

Nicolay Avilov, Mar 01 2024

Keywords

• nonn

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

Examples

			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.
		

Links

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

Crossrefs

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

Programs

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

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.