A344410 -id:A344410 - cp's OEIS frontend

A027669 Numbers k such that for some m, the sum of the first m k-gonal numbers is again a k-gonal number.

3, 4, 6, 8, 10, 11, 14, 17, 20, 23, 26, 29, 30, 32, 35, 38, 41, 43, 44, 47, 50, 53, 56, 59, 60, 62, 65, 68, 71, 74, 77, 80, 83, 86, 88, 89, 92, 95, 98, 101, 104, 107, 110, 113, 116, 119, 122, 125, 128, 131, 134, 137, 140, 143, 145, 146, 149, 152, 155, 158, 161, 164, 167, 170, 173, 176, 179, 182, 185, 188, 191, 194, 197, 200, 203, 206, 209, 212, 215, 218, 221, 224, 227, 230, 233, 236, 239, 242, 245, 248, 251, 254, 257, 260, 263, 266, 269, 272, 275, 276

Offset: 1

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

nonn

The m-th k-gonal number is m+(k-2)*(m^2-m)/2.

3*j+2 = A016789(j) is a term for j >= 2.

Cf. A016789, A027696, A344410.

Set union of A027696 and A016789, excluding elements 2 and 5. - Max Alekseyev, Feb 27 2025

More terms from Max Alekseyev, Feb 27 2025

A373711 Numbers that are simultaneously k-gonal and k-gonal pyramidal for some k >= 3.

Original entry on oeis.org

0, 1, 10, 120, 175, 441, 946, 1045, 1540, 4900, 5985, 7140, 23001, 23725, 48280, 195661, 245905, 314755, 801801, 975061, 1169686, 3578401, 10680265, 27453385, 55202400, 63016921, 101337426, 132361021, 197427385, 258815701, 432684460, 477132085, 837244045

Offset: 1

Kelvin Voskuijl, Jun 14 2024

nonn

Matt Parker calls these numbers cannonball numbers, after the cannonball problem involving finding a number both square and square pyramidal.

If m==2 (mod 3), the m-gonal number A057145(m,(m^3-6*m^2+3*m+19)/9) = (m^2-4*m-2)*(m^2-4*m+1)*(m^3-6*m^2+3*m+19)/162 = A344410((m-2)/3) is a term. See comment in A027696. - Pontus von Brömssen, Dec 09 2024

			4900 is a term because it is both the 70th square and the 24th square pyramidal number.

Pontus von Brömssen, Table of n, a(n) for n = 1..46
Brady Haran and Matt Parker, 90,525,801,730 Cannon Balls, Numberphile video (2019).
Brady Haran and Matt Parker, Cannon Ball Numbers, Numberphile (2019).
Index to sequences related to polygonal numbers.

Subsequences: A027568, A344376, A344410.

Cf. A027669, A027696, A057145, A080851, A379973, A379974, A379975.

a(n) = A057145(A379973(n),A379974(n)) = A080851(A379973(n)-2,A379975(n)-1). - Pontus von Brömssen, Jan 09 2025

a(13)-a(33) from Pontus von Brömssen, Dec 08 2024

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).

Original entry on oeis.org

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

Offset: 1

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

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

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

Cf. A027669, A057145, A344410, A373711.

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

Name clarified by Max Alekseyev, Feb 27 2025

cp's OEIS Frontend

A027669 Numbers k such that for some m, the sum of the first m k-gonal numbers is again a k-gonal number.

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A373711 Numbers that are simultaneously k-gonal and k-gonal pyramidal for some k >= 3.

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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).

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