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

0, 30, 91, 140, 330, 630, 840, 1540, 3900, 6090, 6930, 9180, 9455, 9750, 10920, 18564, 22770, 42900, 46200, 56730, 56925, 58905, 106260, 116886, 128520, 145530, 189420, 191730, 214200, 215600, 242550, 264740, 300300, 308880, 341880, 356265, 364650, 377910, 383180, 399000

Offset: 1

Author

Geoffrey B. Campbell and M. F. Hasler, Jul 03 2018

Keywords

• nonn

Comments

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).

The initial a(1) = 0 is added for completeness.

Below 10^8, only the two numbers 2748900 and 5705700 have three representations of the given form, and none has four.

Links

• Geoffrey Campbell, 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, Number Theory group on LinkedIn, June 2018.

Crossrefs

Cf. A000330, A000384, A306122.

Programs

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}