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

1,1

Comments

New terms should not be added to the Data unless there is a proof that the intervening numbers are not members. Other values known to be members can be added in comments such as those below. - Sean A. Irvine, Jun 20 2026

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

This sequence includes 4980, 9325, 9525, 16420, 19605, 31368, 83135, 125070, 210903, 223613, and 191070870. - Ian Olivant, Apr 06 2026; updated by Max Alekseyev, Apr 15 2026

477135, 1301905, 25789368, 38482522, and 401717664 are also terms. - J.W.L. (Jan) Eerland, Jun 18 2026

Links

Table of n, a(n) for n=1..20.

J.W.L. (Jan) Eerland, Papers about this problem, Jun 09 2026.

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

Prog

(Python)
import numpy as np
from collections import Counter
def g(k, m):
    return (k-4+np.sqrt((4*m*(k-2)*(m+1)*(k*m-k-2*m+5))/3+(k-4)*(k-4)))/(2*(k-2))
eligible_k_values = []
for k in range(3, 1000000):
    m_limit_calc = (3*2**53)/(k-2)
    if m_limit_calc < 0:
        continue
    m_upper_bound = int(m_limit_calc**(1/3))
    for m in range(2, m_upper_bound):
        g_k_m_val = g(k, m)
        if np.iscomplex(g_k_m_val):
            continue
        if float(g_k_m_val).is_integer() and (k-2)*g_k_m_val*g_k_m_val < 2**53:
            eligible_k_values.append(k)
k_counts = Counter(eligible_k_values)
printed_k = set()
for k_val in eligible_k_values:
    if k_counts[k_val] > 1 and (k_val + 1) % 3 == 0:
        if k_val not in printed_k:
            printed_k.add(k_val)
    if (k_val + 1) % 3 != 0:
        if k_val not in printed_k:
            printed_k.add(k_val)
sorted_list = sorted(printed_k)
print(sorted_list) # Ian Olivant, Apr 06 2026

Crossrefs

Cf. A027669, A057145, A344410, A373711.

Sequence in context: A184398 A024665 A027669 * A027440 A169866 A029902

Adjacent sequences: A027693 A027694 A027695 * A027697 A027698 A027699

Keyword

nonn,more,changed

Author

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

Extensions

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

Name clarified by Max Alekseyev, Feb 27 2025

Status

approved