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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.

%I A308488 #27 Aug 24 2019 21:43:00
%S A308488 10,4900,0,946,0,1045,0,175,23725,0,0,441,0,0,975061,0,0,3578401,0,0,
%T A308488 10680265,0,0,27453385,0,0,63016921,23001,0,132361021,0,0,258815701,0,
%U A308488 0,477132085,0,0,55202400,0,245905,1408778281,0,0,2286380881,0,0,314755,0,0
%N 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.
%C A308488 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.
%C A308488 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).
%H A308488 James Grime and Brady Haran, <a href="https://www.youtube.com/watch?v=CROeIGfr3gs">The Best Way to Pack Spheres</a>, Numberphile video (2018).
%H A308488 M. Kaneko and K. Tachibana, <a href="https://doi.org/10.1216/rmjm/1030539614">When is a Polygonal Pyramid Number Again Polygonal?</a>, Rocky Mountain Journal of Mathematics, 32 (2002).
%H A308488 Matt Parker and Brady Haran, <a href="https://www.youtube.com/watch?v=q6L06pyt9CA">90,525,801,730 Cannon Balls</a>, Numberphile video (2019).
%o A308488 (PARI) A308488_vec(lim,J=10^6)={my(
%o A308488     pyramid(s,n)=(3*n^2 + n^3*(s-2)-n*(s-5))/6,
%o A308488     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);
%o A308488 vector(lim,s,check(s+2))}
%Y A308488 Cf. A027669, A057145, A080851.
%K A308488 nonn
%O A308488 3,1
%A A308488 _Davis Smith_, Aug 22 2019