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 A125581 Numbers n such that n does not divide the denominator of the n-th harmonic number nor the denominator of the n-th alternating harmonic number. 12
 77, 847, 9317, 102487, 596778, 1127357, 1193556, 6161805, 12323610, 12400927 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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 n-th harmonic number are listed in A074791. Numbers n such that n does not divide the denominator of the n-th 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) LINKS Tanya Khovanova, Non Recursions. Eric Weisstein's World of Mathematics, Harmonic Number. MATHEMATICA f=0; g=0; Do[g=g+1/n; f=f+(-1)^(n+1)/n; If[ !IntegerQ[Denominator[g]/n]&&!IntegerQ[Denominator[f]/n], Print[n]], {n, 1, 10000}] CROSSREFS Cf. A001008, A002805, A003599, A058312, A058313, A074791, A119955, A121594. Sequence in context: A222728 A223649 A205434 * A176632 A093277 A217643 Adjacent sequences:  A125578 A125579 A125580 * A125582 A125583 A125584 KEYWORD hard,more,nonn AUTHOR Alexander Adamchuk, Jan 03 2007 EXTENSIONS More terms from Max Alekseyev, Mar 11 2007 a(8)-a(10) from Max Alekseyev, Mar 19 2007 STATUS approved

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Last modified April 23 01:29 EDT 2021. Contains 343198 sequences. (Running on oeis4.)