If

then

and

….. (1)

….. (2)

(1) – (2) :

since

then

Solutions to are called “Ramanujan Numbers”

The first five of these are:

1729, 4104, 13832 (= 8 × 1729), 20683, 32832 (= 8 × 4104)

The sequence continues with

*39312, 40033, 46683, 64232, 65728, 110656, 110808, 134379, 149389, …*

OEIS A001235 https://oeis.org/A001235

Now, I’m going to use these “Ramanujan Numbers”

to write numbers expressible as the sum of three positive cubes in two different ways

such that

, and

I ignore 13832 and 32832 since they are multiples

Then, I take the first triple pair solutions to

which is the number, 87539319

(1)

(2)

(3)

(1)

(2)

(3)

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Here are the next numbers which are sum of 2 cubes in 3 different ways:

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The lowest number expressible as the sum of two positive cubes in 4 different ways is:

(1)

(2)

(3)

(4)

(5)

(6)

(1)

(2)

(3)

(4)

(5)

(6)

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Here are the next numbers which are sum of 2 cubes in 4 different ways:

*{12625136269928, (12939,21869), (10362,22580), (7068,23066), (4275,23237)}
{21131226514944, (17176,25232), (11772,26916), (8664,27360), (1539,27645)}
{26059452841000, (21930,24940), (14577,28423), (12900,28810), (4170,29620)}
{55707778473984, (26644,33260), (20400,36144), (10872,37896), (4842,38166)}*

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Here are the first few numbers that can be formed by the sum of 2 cubes in 5 different ways:

48988659276962496:

*{(231518,331954),(221424,336588),(205292,342952),(107839,362753),(38787,365757)}*

391909274215699968 :

*{(463036,663908),(442848,673176),(410584,685904),(215678,725506),(77574,731514)}*

490593422681271000 :

*{(579240,666630),(543145,691295),(285120,776070),(233775,781785),(48369,788631)}*

1322693800477987392 :

*{(694554,995862),(664272,1009764),(615876,1028856),(323517,1088259),(116361,1097271)}*

3135274193725599744 :

*{(926072,1327816),(885696,1346352),(821168,1371808),(431356,1451012),(155148,1463028)}*

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