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%I #5 Mar 30 2012 19:01:05
%S 1,393,412,668,932,1096,1425,1676,1706,1959,2258,2476,2590,3819,4162,
%T 4359,4363,4569,4707,5314,5462,5503,5547,5949,6002,6110,6207,6393,
%U 6429,6484,6500,7226,7706,8151,8654,9566,9586,9759,10085,10141,10455,10774
%N Conjectured numbers n such that the trajectory of n as defined in A003508 is unique.
%C The trajectory in A003508, etc., is defined as a(1)=n, for n>1, a(n) = a(n-1) + 1 + sum of distinct prime factors of a(n-1) that are < a(n-1).
%C If n is a term of this sequence then by definition all later terms in the trajectory of n are excluded.
%t a[1] = 1; a[n_] := a[n] = a[n - 1] + 1 + Plus @@ Select[ Flatten[ Table[ #[[1]], {1}] & /@ FactorInteger[ a[n - 1]]], # < a[n - 1] &]; t = Table[ a[n], {n, 1200}]; f[n_] := Module[{b, k = 1}, b[1] = n; b[m_] := b[m] = b[m - 1] + 1 + Plus @@ Select[ Flatten[ Table[ #[[1]], {1}] & /@ FactorInteger[ b[m - 1]]], # < b[m - 1] &]; While[ Position[t, b[k]] == {} && k < 1000, k++ ]; t = Select[ Union[ Join[t, Table[ b[i], {i, 2, k}]]], # > n &]; If[k == 1000, -1, k - 1]]; lst = {1}; Do[ If[ f[n] == -1, AppendTo[lst, n]], {n, 12500}]; lst
%Y Cf. A003508, A105210, A105211, A105212 and A105213.
%K nonn
%O 1,2
%A _R. K. Guy_ and _Robert G. Wilson v_, Apr 14 2005
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