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.

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, 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, 19, 29, 59, 3, 61, 31, 3, 2, 13, 11, 67, 17, 23, 5, 71, 3, 73, 37, 5

Offset: 1

Author

Amiram Eldar, Jul 07 2021

Keywords

• nonn

Comments

First differs from A088387 at n = 30.

First differs from A197861 at n = 24.

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.

In the case of alpha = 1/2, the sequence is A063539 \ {1} whose asymptotic density is g(1/2) = 1 - log(2) (A244009).

Examples

			a(4) = 2 since 1 <= sqrt(4) < 2^2.
a(6) = 3 since 2 <= sqrt(6) < 2*3.
a(30) = 3 since 2 <= sqrt(30) < 2*3*5.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Paul Erdős, Miscellaneous problems in number theory, Proceedings of the Eleventh Manitoba Conference on Numerical Mathematics and Computing (Winnipeg, Man., 1981), Congr. Numer., Vol. 34 (1982), pp. 25-45.

Crossrefs

Cf. A006530, A063539, A088387, A197861, A244009, A346153.

Programs

Mathematica

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]

Formula

a(n) <= A006530(n).

a(p^e) = p for prime p and e>=1.