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

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

Author

Kelvin Voskuijl, Jun 14 2024

Keywords

• nonn

Comments

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

Examples

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

Links

• 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.

Crossrefs

Subsequences: A027568, A344376, A344410.

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

Formula

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

Extensions

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