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a(n) = n for n <= 3. Thereafter a(n) is the least m such that rad(m) = rad(rad(a(n-3)) + rad(a(n-1))) where rad is A007947.
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%I #12 Nov 30 2022 17:26:30

%S 1,2,3,4,8,5,7,9,16,27,6,32,25,11,13,12,17,30,18,23,53,59,82,15,74,78,

%T 93,167,35,64,169,24,128,45,21,529,38,3481,164,60,89,57,87,22,79,166,

%U 94,173,339,433,606,105,538,286,391,929,75,406,1335,90,218,1553

%N a(n) = n for n <= 3. Thereafter a(n) is the least m such that rad(m) = rad(rad(a(n-3)) + rad(a(n-1))) where rad is A007947.

%C Similar sequence to A358093, though here the definition works between a(n-1) and a(n-3) (skipping a(n-2)) so there is no evident parity pattern, and adjacent pairs are not always coprime.

%e To find a(4): a(1)=1, a(3)=3 so a(4) is the least m such that rad(n) = rad(rad(1) + rad(3)) = rad(4) = 2. Since 2 has occurred already, a(4) = 4.

%t nn = 62; c[_] = False; p[_] = q[_] = 1; f[n_] := Times @@ FactorInteger[n][[All, 1]]; Array[Set[{a[#], c[#]}, {#, True}] &, 3]; Array[(q[#]++; p[#]++) &[f[a[#]] ] &, 3]; Set[{i, j, k}, Array[f[a[#]] &, 3]]; Do[s = f[i + k]; If[PrimeQ[s], m = s^p[s]; p[s]++, m = q[s]; While[Nand[! c[m s], PowerMod[s, s, m] == 0], m++]; m *= s]; q[s]++; Set[{a[n], c[m], i, j, k}, {m, True, j, k, s}], {n, 4, nn}]; Array[a, nn] (* _Michael De Vlieger_, Nov 12 2022 *)

%Y Cf. A007947, A358093.

%K nonn

%O 1,2

%A _David James Sycamore_, Nov 11 2022