cp's OEIS Frontend

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).

%I A027696 #33 Feb 28 2025 12:11:14
%S A027696 3,4,6,8,10,11,14,17,30,41,43,50,60,88,145,276,322,374,823,1152
%N 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).
%C A027696 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).
%C A027696 2378, 2386, and 31265 are also terms. See link "Cannon Ball Numbers". - _Pontus von Brömssen_, Jan 08 2025
%C A027696 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
%H A027696 Brady Haran and Matt Parker, <a href="https://www.numberphile.com/cannon-ball-numbers">Cannon Ball Numbers</a>, Numberphile (2019).
%Y A027696 Cf. A027669, A057145, A344410, A373711.
%K A027696 nonn,more
%O A027696 1,1
%A A027696 Masanobu Kaneko (mkaneko(AT)math.kyushu-u.ac.jp)
%E A027696 More terms from Masanobu Kaneko (mkaneko(AT)math.kyushu-u.ac.jp), Jan 05 1998
%E A027696 Name clarified by _Max Alekseyev_, Feb 27 2025