cp's OEIS Frontend

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.

%I A349689 #12 Aug 28 2023 08:20:42
%S A349689 1,24,30,90,96,342,744,812160,330,147258,32784,3314062080,25896,
%T A349689 565632,116412,210317184,145176,6182491392,963108
%N 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.
%C A349689 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.
%e A349689 The elements of the continued fractions of the abundancy index of the terms are:
%e A349689    n        a(n)  elements
%e A349689   --    --------  -------------------------------------
%e A349689    1           1  1
%e A349689    2          24  2,2
%e A349689    3          30  2,2,2
%e A349689    4          90  2,1,1,2
%e A349689    5          96  2,1,1,1,2
%e A349689    6         342  2,3,1,1,3,2
%e A349689    7         744  2,1,1,2,1,1,2
%e A349689    8      812160  3,1,1,1,1,1,1,3
%e A349689    9         330  2,1,1,1,1,1,1,1,2
%e A349689   10      147258  2,3,1,2,5,5,2,1,3,2
%e A349689   11       32784  2,1,1,2,2,1,2,2,1,1,2
%e A349689   12  3314062080  4,2,1,1,2,1,1,2,1,1,2,4
%e A349689   13       25896  2,1,2,1,1,1,2,1,1,1,2,1,2
%e A349689   14      565632  2,1,7,1,1,2,1,1,2,1,1,7,1,2
%e A349689   15      116412  2,2,1,1,1,1,1,8,1,1,1,1,1,2,2
%e A349689   16   210317184  3,1,1,2,3,3,2,1,1,2,3,3,2,1,1,3
%e A349689   17      145176  2,1,1,1,1,1,1,1,4,1,1,1,1,1,1,1,2
%e A349689   18  6182491392  3,1,1,2,7,3,2,2,1,1,2,2,3,7,2,1,1,3
%e A349689   19      963108  2,1,1,1,1,2,1,1,1,3,1,1,1,2,1,1,1,1,2
%t A349689 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]
%o A349689 (PARI) isok(k, n) = my(v=contfrac(sigma(k)/k)); (#v == n) && (v == Vecrev(v));
%o A349689 a(n) = my(k=1); while (!isok(k, n), k++); k; \\ _Michel Marcus_, Nov 25 2021
%Y A349689 Cf. A000203, A017665, A017666, A349685, A349688.
%Y A349689 Similar sequence: A349478.
%K A349689 nonn,more
%O A349689 1,2
%A A349689 _Amiram Eldar_, Nov 25 2021