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User: Henry Nonnemaker

Henry Nonnemaker has authored 1 sequences.

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

Henry Nonnemaker, Jul 02 2023

nonn

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.

			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.

Cf. A005179, A038548, A038549, A000005.

f(n) = min(A005179(2*n-1), A005179(2*n)); \\ A038549
isok(k) = issquare(f(k)); \\ Michel Marcus, Jul 07 2023

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)

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User: Henry Nonnemaker

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

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