A027696 Numbers k >= 2 such that for some m >= 2, the sum of the first m k-gonal numbers is again a k-gonal number, excluding the parametric solution m = (k^2-4*k-2)/3 when k==2 (mod 3).

3, 4, 6, 8, 10, 11, 14, 17, 30, 41, 43, 50, 60, 88, 145, 276, 322, 374, 823, 1152

Offset: 1

Author

Masanobu Kaneko (mkaneko(AT)math.kyushu-u.ac.jp)

Keywords

• nonn
• more

Comments

The parametric solution: if k==2 (mod 3) and k >= 5, the sum of the first (k^2-4*k-2)/3 k-gonal numbers is the ((k^3-6*k^2+3*k+19)/9)-th k-gonal number A057145(k,(k^3-6*k^2+3*k+19)/9) = A344410((k-2)/3).

2378, 2386, and 31265 are also terms. See link "Cannon Ball Numbers". - Pontus von Brömssen, Jan 08 2025

Number k is a term iff the elliptic curve (3*k-6)*y^2 - (3*k-12)*y = (k-2)*x^3 + 3*x^2 - (k-5)*x has an integral point with x >= 2 different from (k^2-4*k-2)/3. The listed values may be incomplete. For example, I was not able to verify that k = 273 is not a term. - Max Alekseyev, Feb 27 2025

Links

• Brady Haran and Matt Parker, Cannon Ball Numbers, Numberphile (2019).

Crossrefs

Cf. A027669, A057145, A344410, A373711.

Extensions

More terms from Masanobu Kaneko (mkaneko(AT)math.kyushu-u.ac.jp), Jan 05 1998

Name clarified by Max Alekseyev, Feb 27 2025