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A349478 a(n) is the least number k such that the sequence of elements of the continued fraction of the harmonic mean of the divisors of k is palindromic with length n, or -1 if no such k exists.

%I A349478 #11 Aug 28 2023 08:20:36
%S A349478 1,15,8,545,21,1131,16,98124,28676,1109305,28672,16837500,1231932,
%T A349478 477021580,6129711,734420331,441972042,4343866215,42741916965,
%U A349478 96692841558,2193739177
%N A349478 a(n) is the least number k such that the sequence of elements of the continued fraction of the harmonic mean of the divisors of k is palindromic with length n, or -1 if no such k exists.
%C A349478 a(23) = 60755428490.
%C A349478 No more terms below 10^11.
%e A349478 The elements of the continued fractions of the harmonic mean of the divisors of the terms are:
%e A349478    n         a(n)   elements
%e A349478   --  -----------   -------------------------------------------
%e A349478    1            1   1
%e A349478    2           15   2,2
%e A349478    3            8   2,7,2
%e A349478    4          545   3,3,3,3
%e A349478    5           21   2,1,1,1,2
%e A349478    6         1131   5,2,1,1,2,5
%e A349478    7           16   2,1,1,2,1,1,2
%e A349478    8        98124   17,1,1,3,3,1,1,17
%e A349478    9        28676   6,1,2,3,1,3,2,1,6
%e A349478   10      1109305   6,1,1,1,1,1,1,1,1,6
%e A349478   11        28672   11,2,1,1,1,10,1,1,1,2,11
%e A349478   12     16837500   24,1,1,1,2,1,1,2,1,1,1,24
%e A349478   13      1231932   18,1,1,1,1,1,8,1,1,1,1,1,18
%e A349478   14    477021580   38,2,3,1,1,1,1,1,1,1,1,3,2,38
%e A349478   15      6129711   14,2,2,1,1,1,1,9,1,1,1,1,2,2,14
%e A349478   16    734420331   20,2,1,1,1,1,1,1,1,1,1,1,1,1,2,20
%e A349478   17    441972042   15,1,3,2,2,1,1,2,15,2,1,1,2,2,3,1,15
%e A349478   18   4343866215   18,1,1,7,1,8,2,1,1,1,1,2,8,1,7,1,1,18
%e A349478   19  42741916965   94,1,1,7,4,1,1,1,1,3,1,1,1,1,4,7,1,1,94
%e A349478   20  96692841558   28,2,4,1,1,4,1,1,1,6,6,1,1,1,4,1,1,4,2,28
%e A349478   21   2193739177   19,1,1,1,3,1,1,1,1,1,9,1,1,1,1,1,3,1,1,1,19
%t A349478 cfhm[n_] := ContinuedFraction[DivisorSigma[0, n]/DivisorSigma[-1, n]]; seq[len_, nmax_] := Module[{s = Table[0, {len}], c = 0, n = 1, i, cf}, While[c < len && n < nmax, cf = cfhm[n]; If[PalindromeQ[cf] && (i = Length[cf]) <= len && s[[i]] == 0, c++; s[[i]] = n]; n++]; TakeWhile[s, # > 0 &]]; seq[11, 10^7]
%Y A349478 Cf. A099377, A099378, A349473, A349477.
%K A349478 nonn,more
%O A349478 1,2
%A A349478 _Amiram Eldar_, Nov 19 2021