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

1,2

Comments

a(23) = 60755428490.

No more terms below 10^11.

Links

Table of n, a(n) for n=1..21.

Example

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

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]

Crossrefs

Cf. A099377, A099378, A349473, A349477.

Sequence in context: A126892 A195035 A245624 * A182165 A139725 A009929

Adjacent sequences: A349475 A349476 A349477 * A349479 A349480 A349481

Keyword

nonn,more

Author

Amiram Eldar, Nov 19 2021

Status

approved