A308488 a(n) is the smallest n-gonal pyramidal number greater than 1 which is also n-gonal; a(n) = 0 when one does not exist.

10, 4900, 0, 946, 0, 1045, 0, 175, 23725, 0, 0, 441, 0, 0, 975061, 0, 0, 3578401, 0, 0, 10680265, 0, 0, 27453385, 0, 0, 63016921, 23001, 0, 132361021, 0, 0, 258815701, 0, 0, 477132085, 0, 0, 55202400, 0, 245905, 1408778281, 0, 0, 2286380881, 0, 0, 314755, 0, 0

Offset: 3

Author

Davis Smith, Aug 22 2019

Keywords

• nonn

Comments

a(n) is the smallest n-gonal number, N, such that, for some m > 1, N is the sum of the first m n-gonal numbers, 0 when one does not exist.

For n > 5, if n == 2 (mod 3), then a(n) > 0 and a(n) <= A080851(n - 2,((n-2)^2)/3 - 3), but there are cases where a(n) > 0 and n !== 2 (mod 3), e.g., a(10).

Links

• James Grime and Brady Haran, The Best Way to Pack Spheres, Numberphile video (2018).
• M. Kaneko and K. Tachibana, When is a Polygonal Pyramid Number Again Polygonal?, Rocky Mountain Journal of Mathematics, 32 (2002).
• Matt Parker and Brady Haran, 90,525,801,730 Cannon Balls, Numberphile video (2019).

Crossrefs

Cf. A027669, A057145, A080851.

Programs

PARI

A308488_vec(lim,J=10^6)={my(
    pyramid(s,n)=(3*n^2 + n^3*(s-2)-n*(s-5))/6,
    check(s)=j=if(lift(Mod(s,3))==2,((s-2)^2)/3-2,J);m=3;while(m<=j,if(ispolygonal(pyramid(s,m),s),return(pyramid(s,m)),m++));0);
vector(lim,s,check(s+2))}