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a(n) is the least index k such that the n-th odd squarefree number A056911(n) divides A110566(k).
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%I #22 Jan 30 2021 04:27:33

%S 1,6,20,42,33,156,20,272,342,2058,506,377,930,77,14406,629,162,1640,

%T 559,2162,4624,1166,110,6498,3422,610,342732,4422,506,4970,5256,42,

%U 6162,6806

%N a(n) is the least index k such that the n-th odd squarefree number A056911(n) divides A110566(k).

%C According to a theorem proven by Shiu (2016), a(n) exists for all n.

%H Peter Shiu, <a href="https://arxiv.org/abs/1607.02863">The denominators of harmonic numbers</a>, arXiv:1607.02863 [math.NT], 2016.

%e n A056911(n) a(n) = k A110566(k)

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

%e 1 1 1 1 = 1 * 1

%e 2 3 6 3 = 3 * 1

%e 3 5 20 15 = 5 * 3

%e 4 7 42 77 = 7 * 11

%e 5 11 33 11 = 11 * 1

%e 6 13 156 13 = 13 * 1

%e 7 15 20 15 = 15 * 1

%e 8 17 272 17 = 17 * 1

%e 9 19 342 931 = 19 * 49

%e 10 21 2058 1911 = 21 * 91

%e 11 23 506 1725 = 23 * 75

%e 12 29 377 319 = 29 * 11

%e 13 31 930 3751 = 31 * 121

%e 14 33 77 33 = 33 * 1

%e 15 35 14406 2430488445 = 35 * 69442527

%e 16 37 629 20313 = 37 * 549

%e 17 39 162 39 = 39 * 1

%e 18 41 1640 6519 = 41 * 159

%e 19 43 559 645 = 43 * 15

%e 20 47 2162 12831 = 47 * 273

%e 21 51 4624 9537 = 51 * 187

%e 22 53 1166 53 = 53 * 1

%e 23 55 110 55 = 55 * 1

%e 24 57 6498 419498967 = 57 * 7359631

%e 25 59 3422 6431 = 59 * 109

%e 26 61 610 41175 = 61 * 675

%e 27 65 342732 974285 = 65 * 14989

%e 28 67 4422 2211 = 67 * 33

%e 29 69 506 1725 = 69 * 25

%e 30 71 4970 2343 = 71 * 33

%e 31 73 5256 7227 = 73 * 99

%e 32 77 42 77 = 77 * 1

%e 33 79 6162 91801713 = 79 * 1162047

%e 34 83 6806 1200097 = 83 * 14459

%t max = 64; osf = Select[Range[1, 64, 2], SquareFreeQ]; m = Length[osf]; c = 0; s = Table[0, {m}]; h = 0; lcm = 1; n = 1; While[c < m, h += 1/n; lcm = LCM[lcm, n]; r = lcm/Denominator[h]; Do[If[s[[k]] == 0 && Divisible[r, osf[[k]]], c++; s[[k]] = n], {k, 1, m}]; n++]; s

%Y Cf. A002805, A003418, A110566, A056911, A112822.

%K nonn,more

%O 1,2

%A _Amiram Eldar_, Jan 29 2021