A349478 - OEIS

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 #11 Aug 28 2023 08:20:36

%S 1,15,8,545,21,1131,16,98124,28676,1109305,28672,16837500,1231932,

%T 477021580,6129711,734420331,441972042,4343866215,42741916965,

%U 96692841558,2193739177

%N 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 a(23) = 60755428490.

%C No more terms below 10^11.

%e The elements of the continued fractions of the harmonic mean of the divisors of the terms are:

%e n a(n) elements

%e -- ----------- -------------------------------------------

%e 1 1 1

%e 2 15 2,2

%e 3 8 2,7,2

%e 4 545 3,3,3,3

%e 5 21 2,1,1,1,2

%e 6 1131 5,2,1,1,2,5

%e 7 16 2,1,1,2,1,1,2

%e 8 98124 17,1,1,3,3,1,1,17

%e 9 28676 6,1,2,3,1,3,2,1,6

%e 10 1109305 6,1,1,1,1,1,1,1,1,6

%e 11 28672 11,2,1,1,1,10,1,1,1,2,11

%e 12 16837500 24,1,1,1,2,1,1,2,1,1,1,24

%e 13 1231932 18,1,1,1,1,1,8,1,1,1,1,1,18

%e 14 477021580 38,2,3,1,1,1,1,1,1,1,1,3,2,38

%e 15 6129711 14,2,2,1,1,1,1,9,1,1,1,1,2,2,14

%e 16 734420331 20,2,1,1,1,1,1,1,1,1,1,1,1,1,2,20

%e 17 441972042 15,1,3,2,2,1,1,2,15,2,1,1,2,2,3,1,15

%e 18 4343866215 18,1,1,7,1,8,2,1,1,1,1,2,8,1,7,1,1,18

%e 19 42741916965 94,1,1,7,4,1,1,1,1,3,1,1,1,1,4,7,1,1,94

%e 20 96692841558 28,2,4,1,1,4,1,1,1,6,6,1,1,1,4,1,1,4,2,28

%e 21 2193739177 19,1,1,1,3,1,1,1,1,1,9,1,1,1,1,1,3,1,1,1,19

%t 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 Cf. A099377, A099378, A349473, A349477.

%K nonn,more

%O 1,2

%A _Amiram Eldar_, Nov 19 2021