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A363981
Integers k such that the smallest integer with k factor pairs has an odd number of divisors.
1
1, 2, 5, 11, 13, 14, 17, 23, 29, 38, 41, 43, 46, 47, 53, 58, 59, 61, 67, 68, 71, 73, 74, 83, 86, 89, 94, 95, 101, 103, 107, 109, 111, 113, 116, 118, 122, 123, 127, 131, 137, 138, 143, 149, 151, 158, 163, 167, 172, 173, 178, 179, 181, 188, 191, 193, 194, 197
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=', ')
(PARI) f(n) = min(A005179(2*n-1), A005179(2*n)); \\ A038549
isok(k) = issquare(f(k)); \\ Michel Marcus, Jul 07 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
Henry Nonnemaker, Jul 02 2023
STATUS
approved