A362613 Number of co-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, 2, 1, 2, 1, 1, 2, 3, 1, 1, 2, 3, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 2, 1, 2, 2, 2, 2

Offset: 1

Author

Gus Wiseman, May 05 2023

Keywords

• nonn

Comments

First differs from A327500 at n = 180.

First differs from A351946 at n = 180.

First differs from A353507 at n = 180.

We define a co-mode in a multiset to be an element that appears at most as many times as each of the others. For example, the co-modes of {a,a,b,b,b,c,c} are {a,c}.

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

Examples

			The factorization of 180 is 2*2*3*3*5, co-modes {5}, so a(180) = 1.
The factorization of 900 is 2*2*3*3*5*5, co-modes {2,3,5}, so a(900) = 3.
The factorization of 8820 is 2*2*3*3*5*7*7, co-modes {5}, so a(8820) = 1.
		

Links

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

Crossrefs

Positions of first appearances are A002110.

Positions of 1's are A359178, counted by A362610.

Positions of terms > 1 are A362606, counted by A362609.

For mode we have A362611, counted by A362614.

Counting partitions by this statistic (co-mode count) gives A362615.

A027746 lists prime factors (with multiplicity).

A112798 lists prime indices, length A001222, sum A056239.

Cf. A000040, A215366, A327473, A327476, A356862, A359908, A362605.

Programs

Mathematica

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

PARI

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

Python

from sympy import factorint
def A362613(n):
    v = factorint(n).values()
    w = min(v,default=0)
    return sum(1 for e in v if e<=w) # Chai Wah Wu, May 08 2023