cp's OEIS Frontend

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

%I A363981 #48 Sep 14 2023 00:53:51
%S A363981 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 A363981 86,89,94,95,101,103,107,109,111,113,116,118,122,123,127,131,137,138,
%U A363981 143,149,151,158,163,167,172,173,178,179,181,188,191,193,194,197
%N A363981 Integers k such that the smallest integer with k factor pairs has an odd number of divisors.
%C A363981 A factor pair of an integer k is an unordered pair of positive integers (a,b) such that a*b=k.
%C A363981 A038549(n) = min(A005179(2n-1), A005179(2n)). This sequence contains values of k where A005179(2k-1) is smaller.
%C A363981 Also values k such that A038549(k) is a perfect square.
%C A363981 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 A363981 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 A363981 (Python)
%o A363981 from sympy.utilities.iterables import multiset_partitions
%o A363981 from sympy.ntheory import factorint, prime
%o A363981 import math
%o A363981 def smallestNumWithNDivisors(n):
%o A363981     partitionsOfPrimeFactors = multiset_partitions(factorint(n, multiple=True))
%o A363981     candidates = []
%o A363981     for partition in partitionsOfPrimeFactors:
%o A363981         factorization = []
%o A363981         for subset in partition:
%o A363981             factorization.append(math.prod(subset))
%o A363981         factorization.sort()
%o A363981         factorization.reverse()
%o A363981         candidate = 1
%o A363981         for j in range(0, len(factorization)):
%o A363981             candidate *= prime(j+1)**(factorization[j]-1)
%o A363981         candidates.append(candidate)
%o A363981     return min(candidates)
%o A363981 for k in range(1,200):
%o A363981     if smallestNumWithNDivisors(2*k-1)<smallestNumWithNDivisors(2*k):
%o A363981         print(k , end=', ')
%o A363981 (PARI) f(n) = min(A005179(2*n-1), A005179(2*n)); \\ A038549
%o A363981 isok(k) = issquare(f(k)); \\ _Michel Marcus_, Jul 07 2023
%Y A363981 Cf. A005179, A038548, A038549, A000005.
%K A363981 nonn
%O A363981 1,2
%A A363981 _Henry Nonnemaker_, Jul 02 2023