A396682 a(n) is twice the area of the convex hull of the cumulative prime-exponent coordinates of integers from 1 to n + 1.

0, 0, 0, 1, 3, 3, 5, 10, 10, 10, 18, 18, 22, 22, 22, 33, 45, 45, 51, 51, 51, 51, 63, 63, 63, 63, 63, 63, 81, 81, 87, 116, 116, 116, 116, 116, 140, 140, 140, 140, 156, 156, 164, 164, 164, 164, 180, 180, 180, 180, 180, 180, 204, 204, 204, 204, 204, 204, 228

Offset

0,5

Comments

For a given n >= 0, we collect the prime factorization components of all numbers up to n + 1. Each prime power p^e maps to a point (p, e) on an integer lattice. This sequence tracks the doubled Euclidean area of the minimal convex polygon enclosing all points accumulated up to that step.

By Pick's Theorem, twice the area of any lattice polygon is always an integer, which removes the half-integers and creates a strictly nondecreasing sequence of integers, thus a(n) = 2 * A(S(n)), where S(n) is the union of points (p, e) for prime powers dividing k for 1 <= k <= n + 1 and A(S(n)) is the calculated area of the convex hull with coordinates (p,e).

Links

Table of n, a(n) for n=0..58.

Example

For n = 3, the range of numbers is 1 to 4. The (prime, exponent) coordinates are: from 2: (2,1); from 3: (3,1); from 4: (2,2). The convex hull of {(2,1), (3,1), (2,2)} is a triangle with vertices at those three points. The area of this triangle is 1/2. Doubling it gives a(3) = 1.
For n = 7, the range of numbers is 1 to 8. The (prime, exponent) coordinates are: from 2: (2,1); from 3: (3,1); from 4: (2,2); from 5: (5,1); from 6: (2,1), (3,1); from 7: (7,1); from 8: (2,3). The convex hull of {(2,1), (3,1), (2,2), (5,1), (2,1), (3,1), (7,1), (2,3)} is a triangle with vertices at (2,1), (7,1) and (2,3). The area of this triangle is 5. Doubling it gives a(7) = 10.

Prog

(Python)
from itertools import chain
from sympy import factorint
from sympy.geometry import convex_hull
def A396682(n): return 0 if n<3 else int(2*convex_hull(*set(chain.from_iterable(factorint(k).items() for k in range(1, n+2)))).area) # Chai Wah Wu, Jun 03 2026

Crossrefs

Cf. A000040, A396642, A396647.

Sequence in context: A132775 A174102 A217521 * A331925 A252943 A294617

Adjacent sequences: A396678 A396679 A396680 * A396683 A396684 A396685

Keyword

nonn

Author

Ctibor O. Zizka, Jun 02 2026

Status

approved