OFFSET
1,2
COMMENTS
A factor pair of an integer k is an unordered pair of positive integers (a,b) such that a*b=k.
A038549(n) = min(A005179(2n-1), A005179(2n)). This sequence contains values of k where A005179(2k-1) is smaller.
Also values k such that A038549(k) is a perfect square.
I do not know if this sequence is infinite or finite, however I have checked integers up to 20000 and continued to find values at a similar density.
EXAMPLE
The smallest number with 5 factor pairs is 36: (1,36), (2,18), (3,12), (4,9), (6,6). 36 has an odd number of divisors, 9. Thus, 5 is a term.
PROG
(Python)
from sympy.utilities.iterables import multiset_partitions
from sympy.ntheory import factorint, prime
import math
def smallestNumWithNDivisors(n):
partitionsOfPrimeFactors = multiset_partitions(factorint(n, multiple=True))
candidates = []
for partition in partitionsOfPrimeFactors:
factorization = []
for subset in partition:
factorization.append(math.prod(subset))
factorization.sort()
factorization.reverse()
candidate = 1
for j in range(0, len(factorization)):
candidate *= prime(j+1)**(factorization[j]-1)
candidates.append(candidate)
return min(candidates)
for k in range(1, 200):
if smallestNumWithNDivisors(2*k-1)<smallestNumWithNDivisors(2*k):
print(k , end=', ')
isok(k) = issquare(f(k)); \\ Michel Marcus, Jul 07 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
Henry Nonnemaker, Jul 02 2023
STATUS
approved