A362611 Number of modes in the prime factorization of n.

0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 2, 2, 1, 2, 2, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 2, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 3, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 2, 1, 1, 2, 2, 2

Offset: 1

Author

Gus Wiseman, May 05 2023

Keywords

• nonn

Comments

A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes of {a,a,b,b,b,c,d,d,d} are {b,d}.

a(n) depends only on the prime signature of n. - Andrew Howroyd, May 08 2023

Examples

			The factorization of 450 is 2*3*3*5*5, modes {3,5}, so a(450) = 2.
The factorization of 900 is 2*2*3*3*5*5, modes {2,3,5}, so a(900) = 3.
The factorization of 1500 is 2*2*3*5*5*5, modes {5}, so a(1500) = 1.
The factorization of 8820 is 2*2*3*3*5*7*7, modes {2,3,7}, so a(8820) = 3.
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..10000

Crossrefs

Positions of first appearances are A002110.

Positions of 1's are A356862, counted by A362608.

Positions of terms > 1 are A362605, counted by A362607.

For co-mode we have A362613, counted by A362615.

This statistic (mode-count) has triangular form A362614.

A027746 lists prime factors (with multiplicity).

A112798 lists prime indices, length A001222, sum A056239.

A359178 ranks partitions with a unique co-mode, counted by A362610.

A362606 ranks partitions with more than one co-mode, counted by A362609.

Cf. A000040, A000720, A001221, A002865, A072774, A215366, A327473, A327476, A359908.

Programs

Mathematica

Table[x=Last/@If[n==1,0,FactorInteger[n]];Count[x,Max@@x],{n,100}]

PARI

a(n) = if(n==1, 0, my(f=factor(n)[,2], m=vecmax(f)); #select(v->v==m, f)) \\ Andrew Howroyd, May 08 2023

Python

from sympy import factorint
def A362611(n): return list(v:=factorint(n).values()).count(max(v,default=0)) # Chai Wah Wu, May 08 2023

Formula

For n > 1, 1 <= a(n) << log n. - Charles R Greathouse IV, May 09 2023

a(n) <= A001221(n), with equality if and only if n is a power of a squarefree number (A072774). - Amiram Eldar, Mar 02 2025