All listed terms except a(1) are conjectural. For the initial terms given, it has been computationally verified that if a greater a(n) exists, it is at least 10000 times larger than the listed value.
Let a number be designated "prime-complete" if, and only if, it has a complete contiguous set of prime divisors from 2 to its greatest prime divisor. The sequence,
A055932, comprises all the prime-complete numbers, together with the number 1.
a(1) = 633555 is the largest term in
A141399 (for the reasons set out there), the sequence of prime-complete products of sequential pairs of integers generated by m*(m+1).
The number of prime-complete numbers m*(m+n), for fixed n, is finite since for higher values of m the prime index of the greatest prime factor, pi(GPF), of the product is larger than the count of distinct prime factors (omega), yet those must be equal for the product to be prime-complete (having no "gaps" in the prime factorization by the Pigeonhole Principle).
Early values, a(2), a(3) and a(4) are multiples of a(1) because their sequences map to the sequence of a(1), as m*(m+n) = n^2*(m/n)*((m/n)+1) which when k=m/n is n^2*k*(k+1) mapping to m*(m+1) when n divides m.
a(5) breaks this pattern since 5*a(1)= 3167775 is not the greatest term of m*(m+5), but rather 3548155 is, because a term for k*(k+1) at k=709631, that is not prime-complete (because it is missing the prime divisor, 5) is supplied with prime divisor, 5, from n^2=5*5. Thus, for some a(n), high termination values are reached above the mapping to a(1) if n "plugs" a gap in the factorization of m*(m+n) by supplying a missing prime factor. Single gap situations are listed in
A391885. An example of double gaps has been found in a(95) where the primes 5 and 19 from 95 fill the gaps in 30944913*30944914. At a(39) and a(78), n supplies the same primes, 3 and 13, to plug double gaps in 8268799*8268800. However, the ability to extend the sequences is always limited because increasing m will, eventually, force the sequence into finite termination by the lower growth rate of the omega of the product.
Other extension cases will happen when n supplies primes sequentially past 19, such as 23. For a large prime n (with prime index >= 14), sequences exhibit 0% divisibility by n, meaning n never divides m. Terminal values in these cases depend on gap arithmetic: some gaps allow high omega (a(199) = 17448375 with omega=9), while others restrict it severely (a(277) = 199650 with omega=6), creating non-monotonic behavior.
The source code in C is available, at the attached link, to search the m*(m+n) sequences, given n, or a range of n values.
