cp's OEIS Frontend

A306122 Numbers that are product of a second hexagonal number (A014105) and a square pyramidal numbers (A000330) in at least two ways.

%I A306122 #17 Jul 10 2018 10:12:01
%S A306122 0,105,300,855,1155,2940,13860,14700,17850,20790,22230,27300,33930,
%T A306122 70125,73920,87780,114400,116025,135135,145530,157080,195000,213150,
%U A306122 235290,304590,347655,381150,431340,451044,471975,566580,632700,764400,796950,942480,950040
%N A306122 Numbers that are product of a second hexagonal number (A014105) and a square pyramidal numbers (A000330) in at least two ways.
%C A306122 We have A000330(n) = 1 + 2^2 + ... + n^2 and A014105(m) = 0^2 - 1^2 + 2^2 -+ ... + (2m)^2, so the terms of this sequence are the numbers that are a product, in at least two ways, of a partial sum of squares times a (positive) partial sum of squares with alternating signs (with + for even terms; cf. A306121 for the opposite convention).
%C A306122 The initial a(1) = 0 is added for completeness.
%C A306122 Below 10^8, the number 17850 is the only one to have four representations of the given form, and 6347250 is the only one to have exactly three.
%H A306122 Geoffrey Campbell, <a href="https://www.linkedin.com/groups/4510047/4510047-6417378003176783876">Integer solutions of (1²-2²+3²-...+(2a-1)²) × (1²+2²+3²+...+b²) = (1²-2²+3²-...+(2c-1)²) × (1²+2²+3²+...+d²) where a ≠ c and b ≠ d</a>, Number Theory group on LinkedIn, June 2018.
%o A306122 (PARI) {my(L=10^6,A14105(a)=a*(2*a+1),A330(b)=(b+1)*b*(2*b+1)/6,A=S=[]); for(b=1, sqrtnint(L\A14105(1)\3,3), for(a=1,oo, if( setsearch(S,t=A14105(a)*A330(b)), A=setunion(A,[t]), t>L&&next(2); S=setunion(S,[t]))));A}
%Y A306122 Cf. A000330, A014105, A306121.
%K A306122 nonn
%O A306122 1,2
%A A306122 Geoffrey B. Campbell and _M. F. Hasler_, Jul 03 2018