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A346152 a(n) is the least prime divisor p_j of n such that if n = Product_{i=1..k} p_i^e_i and p_1 < p_2 < ... < p_k, then Product_{i=1..j-1} p_i^e_i <= sqrt(n) < Product_{i=j..k} p_i^e_i. a(1) = 1.

%I A346152 #10 Oct 15 2023 05:12:48
%S A346152 1,2,3,2,5,3,7,2,3,5,11,2,13,7,5,2,17,3,19,5,7,11,23,2,5,13,3,7,29,3,
%T A346152 31,2,11,17,7,3,37,19,13,2,41,7,43,11,3,23,47,2,7,5,17,13,53,3,11,2,
%U A346152 19,29,59,3,61,31,3,2,13,11,67,17,23,5,71,3,73,37,5
%N A346152 a(n) is the least prime divisor p_j of n such that if n = Product_{i=1..k} p_i^e_i and p_1 < p_2 < ... < p_k, then Product_{i=1..j-1} p_i^e_i <= sqrt(n) < Product_{i=j..k} p_i^e_i. a(1) = 1.
%C A346152 First differs from A088387 at n = 30.
%C A346152 First differs from A197861 at n = 24.
%C A346152 Erdős (1982) proved that for any 0 <= alpha <= 1, the asymptotic density g(alpha) of numbers k with a(k) < k^alpha exists, and that it is continuous and strictly increasing between g(0) = 0 and g(1) = 1.
%C A346152 In the case of alpha = 1/2, the sequence is A063539 \ {1} whose asymptotic density is g(1/2) = 1 - log(2) (A244009).
%H A346152 Amiram Eldar, <a href="/A346152/b346152.txt">Table of n, a(n) for n = 1..10000</a>
%H A346152 Paul Erdős, <a href="https://users.renyi.hu/~p_erdos/1982-08.pdf">Miscellaneous problems in number theory</a>, Proceedings of the Eleventh Manitoba Conference on Numerical Mathematics and Computing (Winnipeg, Man., 1981), Congr. Numer., Vol. 34 (1982), pp. 25-45.
%F A346152 a(n) <= A006530(n).
%F A346152 a(p^e) = p for prime p and e>=1.
%e A346152 a(4) = 2 since 1 <= sqrt(4) < 2^2.
%e A346152 a(6) = 3 since 2 <= sqrt(6) < 2*3.
%e A346152 a(30) = 3 since 2 <= sqrt(30) < 2*3*5.
%t A346152 a[1] = 1; a[n_] := Module[{fct = FactorInteger[n], prods, ind}, prods = Rest @ FoldList[Times, 1, Power @@@ fct]; ind = FirstPosition[prods^2, _?(# > n &)][[1]]; fct[[ind, 1]]]; Array[a, 100]
%Y A346152 Cf. A006530, A063539, A088387, A197861, A244009, A346153.
%K A346152 nonn
%O A346152 1,2
%A A346152 _Amiram Eldar_, Jul 07 2021