

A295369


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


4




OFFSET

1,4


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.


LINKS



EXAMPLE

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.


PROG

(PARI)
my(centerm = sigmam/(2*msigmam), s=0);
if (n==1,
if (centerm > p, primepi(ceil(centerm)1)  primepi(p), 0),
p = max(floor(centerm), p); while (0<c=A295369(n1, p=nextprime(p+1), m*p, sigmam*(p+1)), s+=c); s
)
}
(SAGE)
def A295369(n, p=1, m=1, sigmam=1):
centerm = sigmam/(2*msigmam)
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(n1, p, m*p, sigmam*(p+1))
if c <= 0: return s
s+=c


CROSSREFS



KEYWORD

nonn,hard,more


AUTHOR



STATUS

approved



