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

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

Table of n, a(n) for n=1..7.

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.

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

Adjacent sequences: A298154 A298155 A298156 * A298158 A298159 A298160

Keyword

nonn,more,hard

Author

Gianluca Amato, Feb 15 2018

Status

approved