A363981 - OEIS

%I #48 Sep 14 2023 00:53:51

%S 1,2,5,11,13,14,17,23,29,38,41,43,46,47,53,58,59,61,67,68,71,73,74,83,

%T 86,89,94,95,101,103,107,109,111,113,116,118,122,123,127,131,137,138,

%U 143,149,151,158,163,167,172,173,178,179,181,188,191,193,194,197

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

%C A factor pair of an integer k is an unordered pair of positive integers (a,b) such that a*b=k.

%C A038549(n) = min(A005179(2n-1), A005179(2n)). This sequence contains values of k where A005179(2k-1) is smaller.

%C Also values k such that A038549(k) is a perfect square.

%C 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.

%e 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.

%o (Python)

%o from sympy.utilities.iterables import multiset_partitions

%o from sympy.ntheory import factorint, prime

%o import math

%o def smallestNumWithNDivisors(n):

%o     partitionsOfPrimeFactors = multiset_partitions(factorint(n, multiple=True))

%o     candidates = []

%o     for partition in partitionsOfPrimeFactors:

%o         factorization = []

%o         for subset in partition:

%o             factorization.append(math.prod(subset))

%o         factorization.sort()

%o         factorization.reverse()

%o         candidate = 1

%o         for j in range(0, len(factorization)):

%o             candidate *= prime(j+1)**(factorization[j]-1)

%o         candidates.append(candidate)

%o     return min(candidates)

%o for k in range(1,200):

%o     if smallestNumWithNDivisors(2*k-1)<smallestNumWithNDivisors(2*k):

%o         print(k , end=', ')

%o (PARI) f(n) = min(A005179(2*n-1), A005179(2*n)); \\ A038549

%o isok(k) = issquare(f(k)); \\ _Michel Marcus_, Jul 07 2023

%Y Cf. A005179, A038548, A038549, A000005.

%K nonn

%O 1,2

%A _Henry Nonnemaker_, Jul 02 2023