A298157 Number of primitive abundant numbers (A071395) with n prime factors, counted with multiplicity.

0, 0, 2, 25, 906, 265602, 13232731828

Offset: 1

Author

Gianluca Amato, Feb 15 2018

Keywords

• nonn
• more
• hard

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).

Examples

			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.
		

Links

• G. Amato, Primitive Weirds and Abundant Numbers, GitHub.
• Gianluca Amato, Maximilian F. Hasler, Giuseppe Melfi, Maurizio Parton, Primitive abundant and weird numbers with many prime factors, arXiv:1802.07178 [math.NT], 2018.

Crossrefs

Cf. A071395 (primitive abundant numbers), A091191 (alternative definition), A287728 (counts of odd PAN), A295369 (counts of squarefree PAN).

Programs

SageMath

# See GitHub link.