A356862 Numbers with a unique largest prime exponent.

2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 24, 25, 27, 28, 29, 31, 32, 37, 40, 41, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 60, 61, 63, 64, 67, 68, 71, 72, 73, 75, 76, 79, 80, 81, 83, 84, 88, 89, 90, 92, 96, 97, 98, 99, 101, 103, 104

Offset

1,1

Comments

If the prime factorization of k has a unique largest exponent, then k is a term.

Numbers whose multiset of prime factors (with multiplicity) has a unique mode. - Gus Wiseman, May 12 2023

Links

Jens Ahlström, Table of n, a(n) for n = 1..10000

Wikipedia, Mode (statistics).

Example

Prime powers (A246655) are in the sequence, since they have only one prime exponent in their prime factorization, hence a unique largest exponent.
144 is in the sequence, since 144 = 2^4 * 3^2 and there is the unique largest exponent 4.
225 is not in the sequence, since 225 = 3^2 * 5^2 and the largest exponent 2 is not unique, but rather it is the exponent of both the prime factor 3 and of the prime factor 5.

Mathematica

Select[Range[2, 100], Count[(e = FactorInteger[#][[;; , 2]]), Max[e]] == 1 &] (* Amiram Eldar, Sep 01 2022 *)

Prog

(Python)
from sympy import factorint
from collections import Counter
def ok(k):
    c = Counter(factorint(k)).most_common(2)
    return not (len(c) > 1 and c[0][1] == c[1][1])
print([k for k in range(2, 105) if ok(k)])
(Python)
from sympy import factorint
from itertools import count, islice
def A356862_gen(startvalue=2): # generator of terms >= startvalue
    return filter(lambda n:len(f:=sorted(factorint(n).values(), reverse=True))==1 or f[0]!=f[1], count(max(startvalue, 2)))
A356862_list = list(islice(A356862_gen(), 30)) # Chai Wah Wu, Sep 10 2022
(PARI) isok(k) = if (k>1, my(f=factor(k), m=vecmax(f[, 2]), w=select(x->(f[x, 2] == m), [1..#f~])); #w == 1); \\ Michel Marcus, Sep 01 2022

Crossrefs

Subsequence of A319161 (which has additional terms 1, 180, 252, 300, 396, 450, 468, ...).

Cf. A246655, A356838, A356840.

For factors instead of exponents we have A102750.

For smallest instead of largest we have A359178, counted by A362610.

The complement is A362605, counted by A362607.

The complement for co-mode is A362606, counted by A362609.

Partitions of this type are counted by A362608.

These are the positions of 1's in A362611, for co-modes A362613.

A001221 is the number of prime exponents, sum A001222.

A027746 lists prime factors, A112798 indices, A124010 exponents.

A362614 counts partitions by number of modes, A362615 co-modes.

Cf. A000041, A002865, A051903, A056239, A070003, A247180, A283050, A327473.

Sequence in context: A319161 A325370 A329139 * A351294 A130091 A359178

Adjacent sequences: A356859 A356860 A356861 * A356863 A356864 A356865

Keyword

nonn,easy

Author

Jens Ahlström, Sep 01 2022

Status

approved