cp's OEIS Frontend

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

%I A295369 #40 Dec 12 2023 08:28:22
%S A295369 0,0,1,18,610,216054,12566567699
%N A295369 Number of squarefree primitive abundant numbers (A071395) with n prime factors.
%C A295369 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 A295369 See A287590 for the number of squarefree ODD primitive abundant numbers with n prime factors.
%C A295369 The actual numbers are listed in A298973. - _M. F. Hasler_, Feb 16 2018
%H A295369 Gianluca Amato, <a href="https://github.com/amato-gianluca/weirds">Primitive Weirds and Abundant Numbers</a>, GitHub.
%H A295369 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 A295369 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 A295369 For n=4, the 18 SFPAN range from 2*5*11*13 = 1430 to 2*5*11*53 = 5830.
%e A295369 For n=5, the 610 SFPAN range from 3*5*7*11*13 = 15015 to 2*5*11*59*647 = 4199030.
%o A295369 (PARI)
%o A295369 A295369(n, p=1, m=1, sigmam=1) = {
%o A295369   my(centerm = sigmam/(2*m-sigmam), s=0);
%o A295369   if (n==1,
%o A295369     if (centerm > p, primepi(ceil(centerm)-1) - primepi(p), 0),
%o A295369     p = max(floor(centerm),p); while (0<c=A295369(n-1, p=nextprime(p+1), m*p, sigmam*(p+1)), s+=c); s
%o A295369   )
%o A295369 }
%o A295369 (SageMath)
%o A295369 def A295369(n, p=1, m=1, sigmam=1):
%o A295369   centerm = sigmam/(2*m-sigmam)
%o A295369   if n==1:
%o A295369     return prime_pi(ceil(centerm)-1) - prime_pi(p) if centerm > p else 0
%o A295369   else:
%o A295369     p = max(floor(centerm), p)
%o A295369     s = 0
%o A295369     while True:
%o A295369        p = next_prime(p)
%o A295369        c = A295369(n-1, p, m*p, sigmam*(p+1))
%o A295369        if c <= 0: return s
%o A295369        s+=c
%Y A295369 Cf. A071395 (primitive abundant numbers), A287590 (counts of odd SFPAN), A298973, A249242 (using A091191).
%K A295369 nonn,hard,more
%O A295369 1,4
%A A295369 _Gianluca Amato_, Feb 12 2018