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 A076661 Index of first term of the harmonic sequence having the same denominator as the partial harmonic sequence beginning with 1/n. 0

%I #4 Mar 31 2012 21:03:40

%S 1,2,4,9,9,10,10,14,25,27,27,27,27,27,27,27,49,49,49,49,49,49,49,49,

%T 49,50,50,125,125,125,125,125,125,125,143,143,143,143,143,136,136,136,

%U 136,136,136,136,136,136,136,136,98,98,98,133,133,133,133,125,125,125,125

%N Index of first term of the harmonic sequence having the same denominator as the partial harmonic sequence beginning with 1/n.

%C Of more interest is the index of terms after which the denominators of the harmonic sequence always match the denominators of the partial harmonic sequence. Notice that 1/4+..1/21 has denominator 15519504, but 1/1+1/2+..1/21 has denominator 5173168.

%e a(4) = firstHM[4] = 9 because 1/4+1/5+1/6+1/7+1/8+1/9 has the same denominator (2520) as 1/1+1/2+..+1/8+1/9 (and the sums to 4,5,6,7 and 8 do not).

%t harmNumber[m_, n_] := HarmonicNumber[n] - HarmonicNumber[m - 1]; denH[n_] := Denominator[HarmonicNumber[n]]; denH[m_, n_] := Denominator[harmNumber[m, n]]; firstHM[m_] := Do[If[denH[k] == denH[m, k], Return[k], ], {k, m, 10^4}]

%Y Cf. A002805.

%K nonn

%O 1,2

%A _Hollie L. Buchanan II_, Oct 24 2002

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