### A011541 - Taxi-cab (taxicab) or Hardy-Ramanujan numbers: the smallest number that is the sum of 2 cubes in n ways

A011541 - Taxi-cab (taxicab) or Hardy-Ramanujan numbers: the smallest number that is the sum of 2 cubes in n ways

1729, 4104, 13832, 20683, 32832, 39312, …

Srinivasa Ramanujan was an Indian mathematician born in Tamil Nadu in 1887. As a child, he showed quick growth in mathematical and analytic abilities. He mastered advanced trigonometry at 12, and invented his own method to solve quartic equations at 15. Two years later, he was calculating Euler’s constant and studying Bernoulli numbers. He struggled through his twenties to keep university scholarships or a job (he was too focused on math).

In 1913, at the age of 25, he sent a letters to several members of faculty at the University of Cambridge. Only G.H. Hardy recognized his brilliance, and invited him to visit and work with him. For the next several years, it became clear that although Ramanujan’s mind would jump to a solution to a problem, he often struggled to prove that the solution was correct. Nonetheless, his work expanded the fields of mathematical analysis, number theory, infinite series and continued fractions.

In 1917, Ramanujan fell seriously ill and he recovered in a nursing home for at least a year. He recovered enough by early 1919 to travel back to India, where he fell ill again and died at the tragic age of 32. His singular mind for math is perhaps best represented by a story his friend and mentor Hardy told about a visit while Ramanujan was convalescing:

Once, in a taxi from London, Hardy noticed its number, 1729. He must have thought about it a little because he entered the room where Ramanujan lay in bed and, with scarcely a hello, blurted out his disappointment with it. It was, he declared, “rather a dull number,” adding that he hoped that wasn’t a bad omen. “No, Hardy,” said Ramanujan, “it is a very interesting number. It is the smallest number expressible as the sum of two cubes in two different ways.”

There seems to be disagreement about how to generalize the Hardy-Ramanujan number. See also A001235 - Taxi-cab numbers: sums of 2 cubes in more than 1 way.