A359178 Numbers with a unique smallest 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, 61, 63, 64, 67, 68, 71, 72, 73, 75, 76, 79, 80, 81, 83, 88, 89, 92, 96, 97, 98, 99, 101, 103, 104, 107, 108, 109, 112, 113, 116, 117

Offset: 1

Author

Jens Ahlström, Jan 08 2023

Keywords

• nonn

Comments

180 is the smallest number with a unique smallest prime exponent that is not a member of A130091.

Examples

			2 = 2^1 is a term since it has 1 as a unique smallest exponent.
6 = 2^1 * 3^1 is not a term since it has two primes with the same smallest exponent.
180 = 2^2 * 3^2 * 5^1 is a term since it has 1 as a unique smallest exponent.
		

Links

• Jens Ahlström, Table of n, a(n) for n = 1..9999
• Wikipedia, Mode (statistics).

Crossrefs

For parts instead of multiplicities we have A247180, counted by A002865.

For greatest instead of smallest we have A356862, counted by A362608.

The complement is A362606, counted by A362609.

Partitions of this type are counted by A362610.

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

A001221 counts prime exponents and A001222 adds them up.

A027746 lists prime factors, A112798 indices, A124010 exponents.

Cf. A102750, A130091, A327473, A327476, A359908, A362605, A362615.

Programs

Mathematica

q[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, Count[e, Min[e]] == 1]; Select[Range[2, 200], q] (* Amiram Eldar, Jan 08 2023 *)

PARI

isok(n) = if (n>1, my(f=factor(n), e = vecmin(f[,2])); #select(x->(x==e), f[,2], 1) == 1); \\ Michel Marcus, Jan 27 2023

Python

from sympy import factorint
def ok(k):
  c = sorted(factorint(k).values())
  return len(c) == 1 or c[0] != c[1]
print([k for k in range(2, 118) if ok(k)])

Python

from itertools import count, islice
from sympy import factorint
def A359178_gen(startvalue=2): # generator of terms >= startvalue
    return filter(lambda n:(f:=list(factorint(n).values())).count(min(f))==1,count(max(startvalue,2)))
A359178_list = list(islice(A359178_gen(),20)) # Chai Wah Wu, Feb 08 2023