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

1,1

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.

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.

Example

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.

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));

Crossrefs

Cf. A306213.

Sequence in context: A247402 A339961 A332738 * A184080 A234401 A203726

Adjacent sequences: A359052 A359053 A359054 * A359056 A359057 A359058

Keyword

nonn

Author

Robert Israel, Dec 14 2022

Status

approved