Note that a(1) = 7*11, a(2) = 7*11^2, and a(3) = 7*11^3.
Harmonic numbers are defined as H(n) = Sum_{k=1..n} 1/k = A001008(n)/A002805(n).
Alternating harmonic numbers are defined as H'(n) = Sum_{k=1..n} (1)^(k+1)*1/k = A058313(n)/A058312(n).
Numbers n such that n does not divide the denominator of the nth harmonic number are listed in A074791. Numbers n such that n does not divide the denominator of the nth alternating harmonic number are listed in A121594.
This sequence is the intersection of A074791 and A121594.
Comments from Max Alekseyev, Mar 07 2007: (Start)
While A125581 indeed contains the geometric progression 7*11^n as a subsequence, it also contains other geometric progressions such as: 546*1093^n, 1092*1093^n, 1755*3511^n, 3510*3511^n and 4896*5557^n (see A126196 and A126197). It may also contain some "isolated" terms (i.e. not participating in the geometric progressions) but such terms are harder to find and at the moment I have no proof that they exist.
This is a sketch of my proof that geometric progression 7*11^n and the others mentioned above belong to A125581.
Lemma 1. H'(n) = H(n)  H([n/2]).
Lemma 2. For prime p and integer n >= p, valuation(H(n),p) >= valuation(H([n/p]),p)  1.
Theorem. For an integer b > 1 and a prime number p such that p divides the numerators of both H(b) and H([b/2]), the geometric progression b*p^n belongs to A125581.
Proof. It is enough to show that valuation(H(b*p^n),p) > n and valuation(H'(b*p^n), p) > n. By Lemma 2 we have valuation(H(b*p^n), p) >= valuation(H(b), p)  n >= 1  n > n.
From this inequality and Lemma 1, we have valuation(H'(b*p^n), p) >= min{ valuation(H(b*p^n), p), valuation(H([b*p^n/2]), p) } >= min{ 1  n, valuation(H([b*p^n/2]), p) }. It remains to show that valuation(H([b*p^n/2]), p) >= 1  n.
Again by Lemma 2, we have valuation(H([b*p^n/2]), p) >= valuation(H([b/2]), p)  n >= 1  n, which completes the proof.
It is easy to check that this Theorem holds for the aforementioned geometric progressions. (End)
