A287590 - OEIS

%I #23 May 24 2018 15:13:36

%S 0,0,0,0,87,14172,101053625,3475496953795289

%N Number of squarefree odd primitive abundant numbers with n prime factors.

%C See A287581 for the largest squarefree odd primitive abundant number (A249263) with n prime factors.

%C Squarefree odd primitive abundant numbers (SOPAN) with r prime factors are of the form N = p_1 * ... * p_r with 3 <= p_1 < ... < p_r and such that the abundancy A(p_1 * ... * p_k) < 2 for k < r and > 2 for k = r, where A(N) = sigma(N)/N. For r < 5 this can never be satisfied, the largest possible value is A(3*5*7*11) = 2 - 2/385.

%H Gianluca Amato, Maximilian F. Hasler, Giuseppe Melfi, 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 From _M. F. Hasler_, Jun 26 2017: (Start)

%e All squarefree odd primitive abundant numbers (SOPAN) have at least 5 prime factors, since the abundancy of a product of 4 distinct odd primes cannot be larger than that of N = 3*5*7*11, with A000203(N)/N = 4/3 * 6/5 * 8/7 * 12/11 = 768/385 = 2 - 2/385 < 2.

%e The 87 SOPAN with 5 prime factors range from A249263(1) = 15015 = 3*5*7*11*13 to  A287581(5) = A249263(87) = 442365 = 3*5*7*11*383.

%e The 14172 SOPAN with 6 prime factors range from A188342(6) = A249263(88) = 692835 = 3*5*11*13*17*19 to A287581(6) = 13455037365  = 3*5*7*11*389*29947.

%e The 101053625 SOPAN with 7 prime factors range from A188342(7) = A249263(608) = 22309287 = 3*7*11*13*17*19*23 to A287581(7) = 1725553747427327895 = 3*5*7*11*389*29959*128194559. (End)

%o (PARI) A287590(r,p=2,a=2,s=0,n=precprime(1\(a-1)))={ r>1 || return(primepi(n)-primepi(p)); (p<n && p=n)||n=p; prod(i=1,r,1+1/n=nextprime(n+1))>a && while( 0<n=A287590(r-1,p=nextprime(p+1),a/(1+1/p)),s+=n); s}

%Y Cf. A249263, A287581.

%K nonn,hard,more

%O 1,5

%A _M. F. Hasler_, May 26 2017

%E Added a(8) calculated by Gianluca Amato. - _M. F. Hasler_, Jun 26 2017

%E Example for 101053625 corrected by _Peter Munn_, Jul 23 2017