cp's OEIS Frontend

A306121 Numbers that are product of a hexagonal number (A000384) and a square pyramidal numbers (A000330) in at least two ways.

%I A306121 #10 Jul 03 2018 21:29:02
%S A306121 0,30,91,140,330,630,840,1540,3900,6090,6930,9180,9455,9750,10920,
%T A306121 18564,22770,42900,46200,56730,56925,58905,106260,116886,128520,
%U A306121 145530,189420,191730,214200,215600,242550,264740,300300,308880,341880,356265,364650,377910,383180,399000
%N A306121 Numbers that are product of a hexagonal number (A000384) and a square pyramidal numbers (A000330) in at least two ways.
%C A306121 We have A000330(n) = 1 + 2^2 + ... + n^2 and A000384(m) = 1 - 2^2 + 3^2 -+ ... + (2m-1)^2, so the terms of this sequence are 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 odd terms; cf. A306122 for the opposite convention).
%C A306121 The initial a(1) = 0 is added for completeness.
%C A306121 Below 10^8, only the two numbers 2748900 and 5705700 have three representations of the given form, and none has four.
%H A306121 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 A306121 (PARI) {my(L=10^6,A384(a)=a*(2*a-1),A330(b)=(b+1)*b*(2*b+1)/6,A=S=[]); for(b=1, sqrtnint(L\A384(1)\3,3), for(a=1,oo, if( setsearch(S,t=A384(a)*A330(b)), A=setunion(A,[t]), t>L&&next(2); S=setunion(S,[t]))));A}
%Y A306121 Cf. A000330, A000384, A306122.
%K A306121 nonn
%O A306121 1,2
%A A306121 Geoffrey B. Campbell and _M. F. Hasler_, Jul 03 2018