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COMMENT
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Note that a(1) = 7*11, a(2) = 7*11^2, a(3) = 7*11^3.
Harmonic numbers are defined as H(n) = Sum[ 1/k, {k,1,n} ] = A001008(n)/A002805(n).
Alternating harmonic numbers are defined as H'(n) = Sum[ (-1)^(k+1)*1/k, {k,1,n} ] = A058313(n)/ A058312(n).
Numbers n such that n does not divide the denominator of the n-th harmonic number are listed in A074791(n) = {6, 18, 20, 21, 33, 42, 54, 63, 66, 77, 100, ...}.
Numbers n such that n does not divide the denominator of the n-th alternating harmonic number are listed in A121594(n) = {15, 28, 75, 77, ...}.
Sequence is the intersection of A074791(n) and A121594(n).
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 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) } and we need 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 that completes the proof.
It is easy to check that this Theorem holds for the aforementioned geometric progressions. (End)
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