Pillai sequence

The Pillai sequence is the sequence of integers that have a record number of terms in their greedy representations as sums of prime numbers (and one). It is named after Subbayya Sivasankaranarayana Pillai, who first defined it in 1930.

It would follow from Goldbach's conjecture that every integer greater than one can be represented as a sum of at most three prime numbers. However, finding such a representation could involve solving instances of the subset sum problem, which is computationally difficult. Instead, Pillai considered the following simpler greedy algorithm for finding a representation of ${\displaystyle n}$ as a sum of primes: choose the first prime in the sum to be the largest prime ${\displaystyle p}$ that is at most ${\displaystyle n}$, and then recursively construct the remaining sum recursively for ${\displaystyle n-p}$. If this process reaches zero, it halts. And if it reaches one instead of zero, it must include one in the sum (even though it is not prime), and then halt. For instance, this algorithm represents 122 as 113 + 7 + 2, even though the shorter representations 61 + 61 or 109 + 13 are also possible.

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