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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.
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%I #12 Aug 28 2023 08:20:42

%S 1,24,30,90,96,342,744,812160,330,147258,32784,3314062080,25896,

%T 565632,116412,210317184,145176,6182491392,963108

%N 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 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 The elements of the continued fractions of the abundancy index of the terms are:

%e n a(n) elements

%e -- -------- -------------------------------------

%e 1 1 1

%e 2 24 2,2

%e 3 30 2,2,2

%e 4 90 2,1,1,2

%e 5 96 2,1,1,1,2

%e 6 342 2,3,1,1,3,2

%e 7 744 2,1,1,2,1,1,2

%e 8 812160 3,1,1,1,1,1,1,3

%e 9 330 2,1,1,1,1,1,1,1,2

%e 10 147258 2,3,1,2,5,5,2,1,3,2

%e 11 32784 2,1,1,2,2,1,2,2,1,1,2

%e 12 3314062080 4,2,1,1,2,1,1,2,1,1,2,4

%e 13 25896 2,1,2,1,1,1,2,1,1,1,2,1,2

%e 14 565632 2,1,7,1,1,2,1,1,2,1,1,7,1,2

%e 15 116412 2,2,1,1,1,1,1,8,1,1,1,1,1,2,2

%e 16 210317184 3,1,1,2,3,3,2,1,1,2,3,3,2,1,1,3

%e 17 145176 2,1,1,1,1,1,1,1,4,1,1,1,1,1,1,1,2

%e 18 6182491392 3,1,1,2,7,3,2,2,1,1,2,2,3,7,2,1,1,3

%e 19 963108 2,1,1,1,1,2,1,1,1,3,1,1,1,2,1,1,1,1,2

%t 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 (PARI) isok(k, n) = my(v=contfrac(sigma(k)/k)); (#v == n) && (v == Vecrev(v));

%o a(n) = my(k=1); while (!isok(k, n), k++); k; \\ _Michel Marcus_, Nov 25 2021

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

%Y Similar sequence: A349478.

%K nonn,more

%O 1,2

%A _Amiram Eldar_, Nov 25 2021