cp's OEIS Frontend

A287728 Number of odd primitive abundant numbers with n prime factors, counted with multiplicity.

%I A287728 #35 Dec 12 2023 08:28:26
%S A287728 0,0,0,0,121,15772,102896101,3475842606319962
%N A287728 Number of odd primitive abundant numbers with n prime factors, counted with multiplicity.
%C A287728 There is no odd abundant number (A005231) with less than 5 prime factors counted with multiplicity (cf. A001222).
%C A287728 Sequence A188439 lists the odd primitive abundant numbers (A006038) sorted by increasing number of distinct prime factors. It is known that there are 576 such terms with r = 3 distinct prime factors, but their number for any larger r = omega(x) appears to be unknown as of today.
%C A287728 It appears that a(n) is just slightly larger than A287590(n), the number of squarefree odd primitive abundant numbers (A249263) with n prime factors. Those with a prime factor to a higher power become less frequent because there are increasingly many terms of the form m*p_r where m has abundancy slightly less than 2, and p_r can be any prime between gpf(m) and 1/(2/A(m)-1) which becomes very large as A(m) -> 2. This also makes difficult the computation of a(n) for n >= 8: The lexicographic smallest choice of (p_1,...,p_8) has p_7 = 128194589 and then 128194601 <= p_8 <= 566684450325179, and calculation of primepi(566'684'450'325'179) takes very long.
%H A287728 Gianluca Amato, <a href="https://github.com/amato-gianluca/weirds">Primitive Weirds and Abundant Numbers</a>, GitHub.
%H A287728 Gianluca Amato, Maximilian F. Hasler, Giuseppe Melfi, and Maurizio Parton, <a href="https://arxiv.org/abs/1802.07178">Primitive abundant and weird numbers with many prime factors</a>, arXiv:1802.07178 [math.NT], 2018.
%o A287728 (SageMath) # See GitHub link.
%Y A287728 Cf. A005231, A006038, A188439, A249263, A287590, A001222.
%K A287728 nonn,hard,more
%O A287728 1,5
%A A287728 _M. F. Hasler_, May 30 2017
%E A287728 a(7) from _Gianluca Amato_, Jun 26 2017
%E A287728 a(8) from _Gianluca Amato_, Feb 26 2018