A356838 The smallest of the most common prime factors of n.

2, 3, 2, 5, 2, 7, 2, 3, 2, 11, 2, 13, 2, 3, 2, 17, 3, 19, 2, 3, 2, 23, 2, 5, 2, 3, 2, 29, 2, 31, 2, 3, 2, 5, 2, 37, 2, 3, 2, 41, 2, 43, 2, 3, 2, 47, 2, 7, 5, 3, 2, 53, 3, 5, 2, 3, 2, 59, 2, 61, 2, 3, 2, 5, 2, 67, 2, 3, 2, 71, 2, 73, 2, 5, 2, 7, 2, 79, 2, 3, 2, 83, 2, 5, 2, 3, 2, 89

Offset

2,1

Comments

Pick the prime factors of n with the largest exponent. a(n) is the smallest one of those prime factors.

If the prime factorization of n has a unique largest exponent (A356862), then a(n) = A356840(n). Otherwise a(n) is the smallest of the most common prime factors, while A356840(n) is the largest of them.

a(n) differs from A020639(n) in case the smallest prime factor is not the most common. For example: a(90) = 3, since 90 = 2 * 3^2 * 5 and 3 has the largest exponent. This is different from A020639(90) = 2, since 2 is the smallest prime factor.

Links

Jens Ahlström, Table of n, a(n) for n = 2..9999

Example

a(18) = 3, since 18 = 2 * 3^2 and 3 is the most common prime factor.
a(450) = 3, since 450 = 2 * 3^2 * 5^2 and 3 is the smallest of the most common prime factors.

Mathematica

a[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; p[[FirstPosition[e, Max[e]][[1]]]]]; Array[a, 100, 2] (* Amiram Eldar, Sep 01 2022 *)

Prog

(Python)
from sympy import factorint
from collections import Counter
def a(n):
    return Counter(factorint(n)).most_common(1)[0][0]
(Python)
from sympy import factorint
def A356838(n): return max(factorint(n).items(), key=lambda x:(x[1], -x[0]))[0] # Chai Wah Wu, Sep 10 2022
(PARI) a(n) = my(f=factor(n), m=vecmax(f[, 2]), w=select(x->(f[x, 2] == m), [1..#f~])); vecmin(vector(#w, k, f[w[k], 1])); \\ Michel Marcus, Sep 01 2022

Crossrefs

Cf. A051903, A356840, A020639, A356862.

Sequence in context: A214606 A325643 A353270 * A079879 A071889 A091963

Adjacent sequences: A356835 A356836 A356837 * A356839 A356840 A356841

Keyword

nonn,easy

Author

Jens Ahlström, Aug 31 2022

Status

approved