A178595 Natural numbers sorted by the sum of square roots of prime factors.
1, 2, 3, 5, 7, 4, 6, 11, 9, 13, 10, 15, 14, 17, 8, 19, 21, 25, 12, 22, 23, 18, 35, 26, 33, 20, 27, 49, 39, 30, 29, 28, 34, 55, 31, 16, 45, 38, 42, 65, 51, 50, 77, 24, 37, 57, 63, 44, 75, 46, 91, 36, 70, 85, 41, 52, 66, 40, 69, 43, 95, 54, 105, 121, 98, 125, 78, 119, 99, 60, 58
Offset: 1
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Mathematica
SortBy[Table[{n,Total[N[Sqrt[Flatten[Table[#[[1]],#[[2]]]&/@FactorInteger[n]]]]]},{n,150}],Last][[;;,1]] (* Harvey P. Dale, May 20 2023 *)
Formula
For n >= 1, Sum_{k=1..A001222(a(n))} sqrt(A027746(a(n),k)) < Sum_{k=1..A001222(a(n+1))} sqrt(A027746(a(n+1),k)). - Peter Munn, Aug 17 2022
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