The triple 171078830, 171078831, 171078832 was apparently found by Laurent Hodges and Michael Reid in 1995.
The starting term of the smallest consecutive 4tuple of abundant numbers is at most 141363708067871564084949719820472453374  Bruno Mishutka (bruno.mishutka(AT)googlemail.com), Nov 01 2007
Paul Erdős showed that there are two absolute constants c1, c2 such that for all large n there are at least c1 log log log n but not more than c2 log log log n consecutive abundant numbers less than n.  Bruno Mishutka (bruno.mishutka(AT)googlemail.com), Nov 01 2007
a(n) exists for all n. Proof: since the infinite product Product_{p prime} (1 + 1/p) diverges, we can find a strictly increasing sequence {b(m)} such that b(0) = 0, Product_{k=b(m)+1..b(m+1)} (1 + 1/prime(k)) > 2 for all m. Given n, by Chinese Remainder Theorem, we can find N such that N + m divides Product_{k=b(m)+1..b(m+1)} prime(k) for m = 0..n1, then sigma(N + m)/(N + m) >= Product_{k=b(m)+1..b(m+1)} (1 + 1/prime(k)) > 2.
For example, if N is divisible by 2*3*5, N+1 is divisible by 7*11*...*73, N+2 is divisible by 79*83*...*7499, N+3 is divisible by 7507*7517*...*57081677, N+4 is divisible by 57081679*57081697*...*(some very large prime), then N through N+4 are consecutive abundant numbers.
Of course, the number N found using this method will be extremely large, since Product_{k=1..K} (1 + 1/prime(k)) ~ log(log(K)). (End)
