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

3,1

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

Table of n, a(n) for n=3..52.

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).

Prog

(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))}

Crossrefs

Cf. A027669, A057145, A080851.

Sequence in context: A233250 A273415 A223055 * A180419 A061888 A348082

Adjacent sequences: A308485 A308486 A308487 * A308489 A308490 A308491

Keyword

nonn

Author

Davis Smith, Aug 22 2019

Status

approved