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.

1, 15, 8, 545, 21, 1131, 16, 98124, 28676, 1109305, 28672, 16837500, 1231932, 477021580, 6129711, 734420331, 441972042, 4343866215, 42741916965, 96692841558, 2193739177

Offset: 1

Author

Amiram Eldar, Nov 19 2021

Keywords

• nonn
• more

Comments

a(23) = 60755428490.

No more terms below 10^11.

Examples

			The elements of the continued fractions of the harmonic mean of the divisors of the terms are:
   n         a(n)   elements
  --  -----------   -------------------------------------------
   1            1   1
   2           15   2,2
   3            8   2,7,2
   4          545   3,3,3,3
   5           21   2,1,1,1,2
   6         1131   5,2,1,1,2,5
   7           16   2,1,1,2,1,1,2
   8        98124   17,1,1,3,3,1,1,17
   9        28676   6,1,2,3,1,3,2,1,6
  10      1109305   6,1,1,1,1,1,1,1,1,6
  11        28672   11,2,1,1,1,10,1,1,1,2,11
  12     16837500   24,1,1,1,2,1,1,2,1,1,1,24
  13      1231932   18,1,1,1,1,1,8,1,1,1,1,1,18
  14    477021580   38,2,3,1,1,1,1,1,1,1,1,3,2,38
  15      6129711   14,2,2,1,1,1,1,9,1,1,1,1,2,2,14
  16    734420331   20,2,1,1,1,1,1,1,1,1,1,1,1,1,2,20
  17    441972042   15,1,3,2,2,1,1,2,15,2,1,1,2,2,3,1,15
  18   4343866215   18,1,1,7,1,8,2,1,1,1,1,2,8,1,7,1,1,18
  19  42741916965   94,1,1,7,4,1,1,1,1,3,1,1,1,1,4,7,1,1,94
  20  96692841558   28,2,4,1,1,4,1,1,1,6,6,1,1,1,4,1,1,4,2,28
  21   2193739177   19,1,1,1,3,1,1,1,1,1,9,1,1,1,1,1,3,1,1,1,19
		

Crossrefs

Cf. A099377, A099378, A349473, A349477.

Programs

Mathematica

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]