A197861 -id:A197861 - cp's OEIS frontend

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

Amiram Eldar, Jul 07 2021

nonn

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).

			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.

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.

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

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]

a(n) <= A006530(n).

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

A197862 Prime divisor of n which appears the fewest times previously in the sequence, with ties to the larger prime.

Original entry on oeis.org

2, 3, 2, 5, 3, 7, 2, 3, 5, 11, 3, 13, 7, 5, 2, 17, 3, 19, 5, 7, 11, 23, 2, 5, 13, 3, 7, 29, 5, 31, 2, 11, 17, 7, 3, 37, 19, 13, 5, 41, 7, 43, 11, 5, 23, 47, 2, 7, 2, 17, 13, 53, 3, 11, 7, 19, 29, 59, 5, 61, 31, 7, 2, 13, 11, 67, 17, 23, 7, 71, 3, 73, 37, 5, 19

Offset: 2

Franklin T. Adams-Watters, Oct 18 2011

nonn

Up to n = 100, this differs from the greatest prime factor function A006530 only at n = 24, 48, 50, 80, and 98.

			The only prime divisor of 4 is 2, so a(4) = 2.
The prime divisors of 6 are 2 and 3; in the sequence to that point (2,3,2,5), there are two 2's and 1 3, we take the less common one, so a(6) = 3.
The prime divisors of 12 are 2 and 3; these occur equally often in the sequence to that point, so we take the larger one; a(12)=3.

Cf. A197861, A006530.

al(n)={local(ns=vector(primepi(n)),r=vector(n-1),ps);
  for(k=1,n-1,
    ps=factor(k+1)[,1]~;
    r[k]=ps[1];
    for(j=2,#ps,if(ns[primepi(ps[j])]<=ns[primepi(r[k])],r[k]=ps[j]));
    ns[primepi(r[k])]++);
  r}

cp's OEIS Frontend

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.

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