A376250 -id:A376250 - cp's OEIS frontend

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

Jens Ahlström, Sep 01 2022

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
Disjoint union of A246655 and A376250. The asymptotic density of this sequence, 0.3660366524547281232052..., is equal to the density of A376250 since the prime powers have a zero density. - Amiram Eldar, Sep 17 2024

			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.

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

Subsequence of A319161 (which has additional terms 1, 180, 252, 300, 396, 450, 468, ...).
Cf. A246655, A356838, A356840, A376250.
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.

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

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

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)])

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

A381719 Numbers whose prime indices cannot be partitioned into sets with a common sum.

Original entry on oeis.org

12, 18, 20, 24, 28, 40, 44, 45, 48, 50, 52, 54, 56, 60, 63, 68, 72, 75, 76, 80, 84, 88, 90, 92, 96, 98, 99, 104, 108, 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, 144, 147, 148, 152, 153, 156, 160, 162, 164, 168, 171, 172, 175, 176, 184, 188, 189, 192

Offset: 1

Gus Wiseman, Apr 22 2025

nonn

Differs from A059404, A323055, A376250 in lacking 150.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798, sum A056239.
Also numbers that cannot be factored into squarefree numbers with a common sum of prime indices (A056239).

			The prime indices of 150 are {1,2,3,3}, and {{3},{3},{1,2}} is a partition into sets with a common sum, so 150 is not in the sequence.

Twice-partitions of this type (sets with a common sum) are counted by A279788.
These multiset partitions (sets with a common sum) are ranked by A326534 /\ A302478.
For distinct block-sums we have A381806, counted by A381990 (complement A381992).
For constant blocks we have A381871 (zeros of A381995), counted by A381993.
Partitions of this type are counted by A381994.
These are the zeros of A382080.
Normal multiset partitions of this type are counted by A382429, see A326518.
The complement counted by A383308.
A000041 counts integer partitions, strict A000009.
A001055 counts factorizations, strict A045778.
A050320 counts factorizations into squarefree numbers, see A381078, A381454.
A050326 counts factorizations into distinct squarefree numbers.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A317141 counts coarsenings of prime indices, refinements A300383.
A381633 counts set systems with distinct sums, see A381634, A293243.
Set multipartitions: A089259, A116540, A270995, A296119, A318360.
Cf. A000720, A000961, A001222, A293511, A321455, A358914, A381635, A381717.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
Select[Range[100],Select[mps[prix[#]], SameQ@@Total/@#&&And@@UnsameQ@@@#&]=={}&]

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A356862 Numbers with a unique largest prime exponent.

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