Gianluca Amato - cp's OEIS frontend

User: Gianluca Amato

Gianluca Amato has authored 2 sequences.

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

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

Offset: 1

Gianluca Amato, Feb 15 2018

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).

			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.

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.

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

SageMath
```
# See GitHub link.
```

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

Original entry on oeis.org

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

Offset: 1

Gianluca Amato, Feb 12 2018

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

			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.

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.

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

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
  )
}

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

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User: Gianluca Amato

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

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