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%I #39 Jan 02 2024 09:35:48
%S 0,0,2,25,906,265602,13232731828
%N Number of primitive abundant numbers (A071395) with n prime factors, counted with multiplicity.
%C This uses the first definition of primitive abundant numbers, A071395: having only deficient proper divisors. The second definition (A091191: having no abundant proper divisors) would yield infinite a(3), since all numbers 6*p, p > 3, are in that sequence.
%C See A287728 for the number of ODD primitive abundant numbers with n prime factors, counted with multiplicity and A295369 for the number of squarefree primitive abundant numbers with n distinct prime factors.
%C It appears that a(n) is just slightly larger than A295369(n).
%H G. Amato, <a href="https://github.com/amato-gianluca/weirds">Primitive Weirds and Abundant Numbers</a>, GitHub.
%H Gianluca Amato, Maximilian F. Hasler, Giuseppe Melfi, 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.
%e For n=3, the only two primitive abundant numbers (PAN) are 2*2*5 = 20 and 2*5*7 = 70. The latter is also a primitive weird number, see A002975.
%e For n=4, the 25 PAN range from 2^3*11 = 88 to 2*5*11*53 = 5830.
%o (SageMath) # See GitHub link.
%Y Cf. A071395 (primitive abundant numbers), A091191 (alternative definition), A287728 (counts of odd PAN), A295369 (counts of squarefree PAN).
%K nonn,more,hard
%O 1,3
%A _Gianluca Amato_, Feb 15 2018