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

0, 0, 0, 0, 121, 15772, 102896101, 3475842606319962

Offset: 1

Author

M. F. Hasler, May 30 2017

Keywords

• nonn
• hard
• more

Comments

There is no odd abundant number (A005231) with less than 5 prime factors counted with multiplicity (cf. A001222).

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.

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.

Links

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

Crossrefs

Cf. A005231, A006038, A188439, A249263, A287590, A001222.

Programs

SageMath

# See GitHub link.

Extensions

a(7) from Gianluca Amato, Jun 26 2017

a(8) from Gianluca Amato, Feb 26 2018