A349689 a(n) is the least number k such that the sequence of elements of the abundancy index of k is palindromic with length n, or -1 if no such k exists.

1, 24, 30, 90, 96, 342, 744, 812160, 330, 147258, 32784, 3314062080, 25896, 565632, 116412, 210317184, 145176, 6182491392, 963108

Offset

1,2

Comments

a(21) = 7094832, a(23) = 24167070, a(25) = 858983598, a(27) = 1137635260, a(29) = 1402857468, a(31) = 45230309244, and there are no more terms below 1.6*10^11.

Links

Table of n, a(n) for n=1..19.

Example

The elements of the continued fractions of the abundancy index of the terms are:
   n        a(n)  elements
  --    --------  -------------------------------------
   1           1  1
   2          24  2,2
   3          30  2,2,2
   4          90  2,1,1,2
   5          96  2,1,1,1,2
   6         342  2,3,1,1,3,2
   7         744  2,1,1,2,1,1,2
   8      812160  3,1,1,1,1,1,1,3
   9         330  2,1,1,1,1,1,1,1,2
  10      147258  2,3,1,2,5,5,2,1,3,2
  11       32784  2,1,1,2,2,1,2,2,1,1,2
  12  3314062080  4,2,1,1,2,1,1,2,1,1,2,4
  13       25896  2,1,2,1,1,1,2,1,1,1,2,1,2
  14      565632  2,1,7,1,1,2,1,1,2,1,1,7,1,2
  15      116412  2,2,1,1,1,1,1,8,1,1,1,1,1,2,2
  16   210317184  3,1,1,2,3,3,2,1,1,2,3,3,2,1,1,3
  17      145176  2,1,1,1,1,1,1,1,4,1,1,1,1,1,1,1,2
  18  6182491392  3,1,1,2,7,3,2,2,1,1,2,2,3,7,2,1,1,3
  19      963108  2,1,1,1,1,2,1,1,1,3,1,1,1,2,1,1,1,1,2

Mathematica

cfai[n_] := ContinuedFraction[DivisorSigma[1, n]/n]; seq[len_, nmax_] := Module[{s = Table[0, {len}], c = 0, n = 1, i, cf}, While[c < len && n < nmax, cf = cfai[n]; If[PalindromeQ[cf] && (i = Length[cf]) <= len && s[[i]] == 0, c++; s[[i]] = n]; n++]; TakeWhile[s, # > 0 &]]; seq[11, 10^6]

Prog

(PARI) isok(k, n) = my(v=contfrac(sigma(k)/k)); (#v == n) && (v == Vecrev(v));
a(n) = my(k=1); while (!isok(k, n), k++); k; \\ Michel Marcus, Nov 25 2021

Crossrefs

Cf. A000203, A017665, A017666, A349685, A349688.

Similar sequence: A349478.

Sequence in context: A333122 A135045 A109321 * A348630 A067952 A239801

Adjacent sequences: A349686 A349687 A349688 * A349690 A349691 A349692

Keyword

nonn,more

Author

Amiram Eldar, Nov 25 2021

Status

approved