OFFSET
1,5
COMMENTS
See A287581 for the largest squarefree odd primitive abundant number (A249263) with n prime factors.
Squarefree odd primitive abundant numbers (SOPAN) with r prime factors are of the form N = p_1 * ... * p_r with 3 <= p_1 < ... < p_r and such that the abundancy A(p_1 * ... * p_k) < 2 for k < r and > 2 for k = r, where A(N) = sigma(N)/N. For r < 5 this can never be satisfied, the largest possible value is A(3*5*7*11) = 2 - 2/385.
LINKS
Gianluca Amato, Maximilian F. Hasler, Giuseppe Melfi, Maurizio Parton, Primitive abundant and weird numbers with many prime factors, arXiv:1802.07178 [math.NT], 2018.
EXAMPLE
From M. F. Hasler, Jun 26 2017: (Start)
All squarefree odd primitive abundant numbers (SOPAN) have at least 5 prime factors, since the abundancy of a product of 4 distinct odd primes cannot be larger than that of N = 3*5*7*11, with A000203(N)/N = 4/3 * 6/5 * 8/7 * 12/11 = 768/385 = 2 - 2/385 < 2.
The 87 SOPAN with 5 prime factors range from A249263(1) = 15015 = 3*5*7*11*13 to A287581(5) = A249263(87) = 442365 = 3*5*7*11*383.
PROG
CROSSREFS
KEYWORD
nonn,hard,more
AUTHOR
M. F. Hasler, May 26 2017
EXTENSIONS
Added a(8) calculated by Gianluca Amato. - M. F. Hasler, Jun 26 2017
Example for 101053625 corrected by Peter Munn, Jul 23 2017
STATUS
approved