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A349500 a(n) is the least number k such that A349474(k) = A349474(k+1) = n, or -1 if no such k exists.

%I A349500 #8 Nov 20 2021 20:49:34
%S A349500 2,7,76,56,81,63,913,892,1969,4824,22855,16819,48922,170649,273216,
%T A349500 607783,1204354,1910608,3433671,10104969,19546522,21424744,66961728,
%U A349500 103366113,217458328,813832568,771821712,2370545332,4638470426,7190276806,9309810824,35730615937
%N A349500 a(n) is the least number k such that A349474(k) = A349474(k+1) = n, or -1 if no such k exists.
%C A349500 The sequence begins at n = 2. a(1) != -1 if and only if two consecutive harmonic numbers exist. There are no odd harmonic numbers between 2 and 10^24 (Cohen and Sorli, 2010) and it is conjectured that they do not exist.
%H A349500 Graeme L. Cohen and Ronald M. Sorli, <a href="https://doi.org/10.1090/S0025-5718-10-02337-9">Odd harmonic numbers exceed 10^24</a>, Mathematics of Computation, Vol. 79, No. 272 (2010), pp. 2451-2460.
%e A349500 a(2) = 2 since A349474(2) = A349474(3) = 2 and there is no smaller pair of consecutive numbers with this property.
%e A349500 a(3) = 7 since A349474(7) = A349474(8) = 3 and there is no smaller pair of consecutive numbers with this property.
%t A349500 c[n_] := Length @ ContinuedFraction[DivisorSigma[0, n]/DivisorSigma[-1, n]]; seq[len_, nmax_] := Module[{s = Table[0, {len}], k = 1, n = 1, i}, s[[1]] = -1; While[n < nmax && k < len, i = c[n]; If[c[n+1] == i && i <= len && s[[i]] == 0, k++; s[[i]] = n]; n++]; Rest @ s]; seq[15, 10^6]
%Y A349500 Cf. A001599, A349474, A349499, A349501.
%K A349500 nonn
%O A349500 2,1
%A A349500 _Amiram Eldar_, Nov 20 2021