A295369 - OEIS

%I #40 Dec 12 2023 08:28:22

%S 0,0,1,18,610,216054,12566567699

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

%C 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.

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

%C The actual numbers are listed in A298973. - _M. F. Hasler_, Feb 16 2018

%H Gianluca Amato, <a href="https://github.com/amato-gianluca/weirds">Primitive Weirds and Abundant Numbers</a>, GitHub.

%H Gianluca Amato, Maximilian F. Hasler, Giuseppe Melfi, and Maurizio Parton, <a href="https://arxiv.org/abs/1802.07178">Primitive abundant and weird numbers with many prime factors</a>, arXiv:1802.07178 [math.NT], 2018.

%e For n=3, the only squarefree primitive abundant number (SFPAN) is 2*5*7 = 70, which is also a primitive weird number, see A002975.

%e For n=4, the 18 SFPAN range from 2*5*11*13 = 1430 to 2*5*11*53 = 5830.

%e For n=5, the 610 SFPAN range from 3*5*7*11*13 = 15015 to 2*5*11*59*647 = 4199030.

%o (PARI)

%o A295369(n, p=1, m=1, sigmam=1) = {

%o   my(centerm = sigmam/(2*m-sigmam), s=0);

%o   if (n==1,

%o     if (centerm > p, primepi(ceil(centerm)-1) - primepi(p), 0),

%o     p = max(floor(centerm),p); while (0<c=A295369(n-1, p=nextprime(p+1), m*p, sigmam*(p+1)), s+=c); s

%o   )

%o }

%o (SageMath)

%o def A295369(n, p=1, m=1, sigmam=1):

%o   centerm = sigmam/(2*m-sigmam)

%o   if n==1:

%o     return prime_pi(ceil(centerm)-1) - prime_pi(p) if centerm > p else 0

%o   else:

%o     p = max(floor(centerm), p)

%o     s = 0

%o     while True:

%o        p = next_prime(p)

%o        c = A295369(n-1, p, m*p, sigmam*(p+1))

%o        if c <= 0: return s

%o        s+=c

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

%K nonn,hard,more

%O 1,4

%A _Gianluca Amato_, Feb 12 2018