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%I #9 Nov 27 2021 05:04:20
%S 1,6,24,28,30,120,140,348,496,672,1080,2480,6048,6200,6552,6786,8128,
%T 30240,32760,40640,143880,238080,435708,514080,523776,524160,556920,
%U 805728,1997868,2178540,4713984,23569920,33550336,37035180,38958426,45532800,91963648,142990848
%N Numbers k such that the continued fraction of the abundancy index of k contains a single distinct element.
%C All the multiply-perfect numbers (A007691) are terms of this sequence, since the continued fraction of their abundancy index contains a single element.
%C Up to 4*10^10 the continued fractions of the abundancy indices of the terms have lengths 1, 2, 3, 5 or 11. The least terms that are corresponding to these lengths are 1, 24, 30, 348 and 1997868, respectively. Are there terms with other lengths?
%e 24 is a term since the continued fraction of its abundancy index sigma(24)/24 = 5/2 = 2 + 1/2 has the elements {2, 2}.
%e 30 is a term since the continued fraction of its abundancy index sigma(30)/30 = 12/5 = 2 + 1/(2 + 1/2) has the elements {2, 2, 2}.
%e 143880 is a term since the continued fraction of its abundancy index sigma(143880)/143880 = 360/109 = 3 + 1/(3 + 1/(3 + 1/(3 + 1/3))) has the elements {3, 3, 3}.
%t c[n_] := ContinuedFraction[DivisorSigma[1, n] / n]; q[n_] := Length[Union[c[n]]] == 1; Select[Range[10^6], q]
%o (PARI) isok(k) = #Set(contfrac(sigma(k)/k)) == 1; \\ _Michel Marcus_, Nov 25 2021
%Y Cf. A000203, A007691, A071862, A071865, A017665, A017666, A349685.
%K nonn
%O 1,2
%A _Amiram Eldar_, Nov 25 2021
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