A295369 Number of squarefree primitive abundant numbers (A071395) with n prime factors.

0, 0, 1, 18, 610, 216054, 12566567699

Offset: 1

Author

Gianluca Amato, Feb 12 2018

Keywords

• nonn
• hard
• more

Comments

Here primitive abundant number means an abundant number all of whose proper divisors are deficient numbers (A071395). The alternative definition (an abundant number having no abundant proper divisor, see A091191) would yield an infinite count for a(3): since 2*3 = 6 is perfect, all numbers of the kind 2*3*p with p > 3 would be primitive abundant.

See A287590 for the number of squarefree ODD primitive abundant numbers with n prime factors.

The actual numbers are listed in A298973. - M. F. Hasler, Feb 16 2018

Examples

			For n=3, the only squarefree primitive abundant number (SFPAN) is 2*5*7 = 70, which is also a primitive weird number, see A002975.
For n=4, the 18 SFPAN range from 2*5*11*13 = 1430 to 2*5*11*53 = 5830.
For n=5, the 610 SFPAN range from 3*5*7*11*13 = 15015 to 2*5*11*59*647 = 4199030.
		

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. A071395 (primitive abundant numbers), A287590 (counts of odd SFPAN), A298973, A249242 (using A091191).

Programs

PARI

A295369(n, p=1, m=1, sigmam=1) = {
  my(centerm = sigmam/(2*m-sigmam), s=0);
  if (n==1,
    if (centerm > p, primepi(ceil(centerm)-1) - primepi(p), 0),
    p = max(floor(centerm),p); while (0A295369(n-1, p=nextprime(p+1), m*p, sigmam*(p+1)), s+=c); s
  )
}

SageMath

def A295369(n, p=1, m=1, sigmam=1):
  centerm = sigmam/(2*m-sigmam)
  if n==1:
    return prime_pi(ceil(centerm)-1) - prime_pi(p) if centerm > p else 0
  else:
    p = max(floor(centerm), p)
    s = 0
    while True:
       p = next_prime(p)
       c = A295369(n-1, p, m*p, sigmam*(p+1))
       if c <= 0: return s
       s+=c