[seqfan] Composite such that the number of non-divisors of n divides their sum

Charles Greathouse charles.greathouse at case.edu
Tue Feb 11 20:17:07 CET 2014

Donovan Johnson and Giovanni Resta have completed checks to very large
bounds on A230605, finding no more terms below 4 trillion. A naive
heuristic suggests that there should be infinitely many such numbers, about
log x below x (since almost all numbers are composite, and the divisor is
very close to n).

1. Is this heuristic reasonable, or should we expect different behavior
from this sequence? I think the divisor n - d(n) acts very much like a
random number near n, but the dividend (n*(n+1)/2 - sigma(n) might be
smoother than typical numbers.

2. Is there a better way to search for members of this sequence than using
a factoring sieve and testing numbers individually? It feels like there
might be some way to take advantage of the fact that the divisor is in
[n-k, n-3] for some fairly small k (the d(m), where m is the largest highly
composite number <= n) but I can't find one.

Charles Greathouse
Case Western Reserve University

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