A359612 Largest prime factor with minimal exponent in canonical prime factorization of n.

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

Offset: 2

Author

Jens Ahlström, Jan 06 2023

Keywords

• nonn

Comments

When inspecting the minimal exponent of the canonical representation of n, a(n) is the largest of those primes, while A067695(n) is the smallest.

On the other hand if the maximal exponent is regarded similarly, A356840(n) is the largest of those primes and A356838(n) is the smallest.

18 is the smallest n, where a(n) differs from A006530(n), since a(18) = 2, while A006530(18) = 3.

Examples

			a(162) = a(2^1 * 3^4) = 2.
a(225) = a(3^2 * 5^2) = 5.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 2..20000

Crossrefs

Cf. A006530, A067695, A356838, A356840.

Programs

Maple

a:= n-> (l-> (m-> max(map(i-> i[1], select(y->y[2]=m,
         l))))(min(map(x-> x[2], l))))(ifactors(n)[2]):
seq(a(n), n=2..75);  # Alois P. Heinz, Jan 25 2023

Mathematica

a[n_] := Module[{f = FactorInteger[n], e, ind}, e = f[[;; , 2]]; ind = Position[e, Min[e]][[-1, 1]]; f[[ind, 1]]]; Array[a, 100, 2] (* Amiram Eldar, Jan 07 2023 *)

PARI

a(n) = my(f=factor(n), e=vecmin(f[,2])); f[vecmax(select(x->(x==e), f[,2], 1)), 1]; \\ Michel Marcus, Jan 26 2023

Python

from sympy import factorint
def a(n):
    max_factor = 0
    min_exponent = float("inf")
    for p, exponent in factorint(n).items():
        if exponent < min_exponent:
            max_factor = p
            min_exponent = exponent
        elif exponent == min_exponent:
            max_factor = max(max_factor, p)
    return max_factor

Python

from sympy import factorint
def A359612(n): return (f:=list(map(tuple,zip(*sorted(factorint(n).items(),reverse=True)))))[0][f[1].index(min(f[1]))] # Chai Wah Wu, Feb 07 2023