A359055 Numbers that can be represented in more than one way as the sum of cubes of three distinct positive numbers in arithmetic progression.

5643, 12384, 31977, 45144, 99072, 123849, 152361, 153792, 255816, 259776, 269739, 274968, 334368, 361152, 477576, 500445, 705375, 792576, 863379, 912339, 928017, 950931, 990792, 1090584, 1218888, 1230336, 1548000, 1629144, 1700424, 1737252, 1799523, 1813512, 1935549, 1941192, 2046528, 2078208

Offset: 1

Author

Robert Israel, Dec 14 2022

Keywords

• nonn

Comments

Numbers k such that there are at least two pairs of positive numbers (a,d) such that k = a^3 + (a+d)^3 + (a+2d)^3.

The first term that has three such representations is 255816 = 8^3 + 34^3 + 60^3 = 18^3 + 38^3 + 58^3 = 43^3 + 44^3 + 45^3.

346380489216 has four such representations: 1188^3 + 3888^3 + 6588^3, 1728^3 + 4104^3 + 6480^3, 4248^3 + 4824^3 + 5400^3 and 4665^3 + 4864^3 + 5063^3. It might not be the first.

Examples

			a(1) = 5643 is a term because 5643 = 1^3 + (1+8)^3 + (1+2*8)^3 = 6^3 + (6+5)^3 + (6+2*5)^3.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..2500
• R. Israel et al, Sum of cubes of three positive integers in arithmetic progression in four ways?, Mathematics StackExchange, Dec. 2022.

Crossrefs

Cf. A306213.

Programs

Maple

N:= 10^7: # to get terms <= N
S:= {}: S2:= {}:
for a from 1 while a^3 + (a+1)^3 + (a+2)^3 <= N do
  for d from 1 do
    x:= a^3 + (a+d)^3 + (a+2*d)^3;
    if x > N then break fi;
    if member(x,S) then S2:= S2 union {x} fi;
    S:= S union {x}
od od:
sort(convert(S,list));