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A298157
Number of primitive abundant numbers (A071395) with n prime factors, counted with multiplicity.
0
0, 0, 2, 25, 906, 265602, 13232731828
OFFSET
1,3
COMMENTS
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.
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.
It appears that a(n) is just slightly larger than A295369(n).
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
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.
For n=4, the 25 PAN range from 2^3*11 = 88 to 2*5*11*53 = 5830.
PROG
(SageMath) # See GitHub link.
CROSSREFS
Cf. A071395 (primitive abundant numbers), A091191 (alternative definition), A287728 (counts of odd PAN), A295369 (counts of squarefree PAN).
Sequence in context: A059363 A216661 A301616 * A342298 A273545 A014050
KEYWORD
nonn,more,hard
AUTHOR
Gianluca Amato, Feb 15 2018
STATUS
approved