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A370759
Numbers expressible in the form k*m + 2*(k+m) - 1, for positive k and m.
1
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
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
KEYWORD
nonn
AUTHOR
Nicolay Avilov, Mar 01 2024
STATUS
approved