A363981 Integers k such that the smallest integer with k factor pairs has an odd number of divisors.

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.

Links

Table of n, a(n) for n=1..58.

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

Cf. A005179, A038548, A038549, A000005.

Sequence in context: A062889 A065433 A032541 * A281644 A075749 A138837

Adjacent sequences: A363978 A363979 A363980 * A363982 A363983 A363984

Keyword

nonn

Author

Henry Nonnemaker, Jul 02 2023

Status

approved