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

0, 105, 300, 855, 1155, 2940, 13860, 14700, 17850, 20790, 22230, 27300, 33930, 70125, 73920, 87780, 114400, 116025, 135135, 145530, 157080, 195000, 213150, 235290, 304590, 347655, 381150, 431340, 451044, 471975, 566580, 632700, 764400, 796950, 942480, 950040

Offset

1,2

Comments

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).

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

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.

Links

Table of n, a(n) for n=1..36.

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.

Prog

(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}

Crossrefs

Cf. A000330, A014105, A306121.

Sequence in context: A075764 A046299 A010090 * A174830 A325312 A147576

Adjacent sequences: A306119 A306120 A306121 * A306123 A306124 A306125

Keyword

nonn

Author

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

Status

approved