A349477 -id:A349477 - cp's OEIS frontend

A349688 Numbers k such that the sequence of elements of the continued fraction of the abundancy index of k is palindromic.

1, 6, 24, 28, 30, 42, 54, 66, 70, 78, 84, 90, 96, 102, 114, 120, 138, 140, 174, 186, 220, 222, 246, 258, 264, 270, 282, 308, 318, 330, 342, 348, 354, 364, 366, 402, 426, 438, 474, 476, 496, 498, 532, 534, 582, 606, 618, 642, 644, 654, 660, 672, 678, 744, 760, 762

Offset: 1

Amiram Eldar, Nov 25 2021

nonn

All the multiply-perfect numbers (A007691) are terms of this sequence, since the continued fraction of their abundancy index contains a single element.

			24 is a term since the sequence of elements of the abundancy index of 24, sigma(24)/24 = 5/2 = 2 + 1/2, is {2, 2}, which is palindromic.
42 is a term since the sequence of elements of the abundancy index of 42, sigma(42)/42 = 16/7 = 2 + 1/(3 + 1/2), is {2, 3, 2}, which is palindromic.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A349685.
A007691 and A349686 are subsequences.
Similar sequence: A349477.

q[n_] := PalindromeQ[ContinuedFraction[DivisorSigma[1, n]/n]]; Select[Range[1000], q]

isok(k) = my(v=contfrac(sigma(k)/k)); v == Vecrev(v); \\ Michel Marcus, Nov 25 2021

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.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Nov 19 2021

a(23) = 60755428490.

No more terms below 10^11.

			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

Cf. A099377, A099378, A349473, A349477.

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]

cp's OEIS Frontend

A349688 Numbers k such that the sequence of elements of the continued fraction of the abundancy index of k is palindromic.

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