A287590 Number of squarefree odd primitive abundant numbers with n prime factors.

0, 0, 0, 0, 87, 14172, 101053625, 3475496953795289

Offset

1,5

Comments

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

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.

Links

Table of n, a(n) for n=1..8.

Gianluca Amato, Maximilian F. Hasler, Giuseppe Melfi, Maurizio Parton, Primitive abundant and weird numbers with many prime factors, arXiv:1802.07178 [math.NT], 2018.

Example

From M. F. Hasler, Jun 26 2017: (Start)
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.
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.
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.
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)

Prog

(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}

Crossrefs

Cf. A249263, A287581.

Sequence in context: A291130 A183040 A072692 * A133391 A298832 A298620

Adjacent sequences: A287587 A287588 A287589 * A287591 A287592 A287593

Keyword

nonn,hard,more

Author

M. F. Hasler, May 26 2017

Extensions

Added a(8) calculated by Gianluca Amato. - M. F. Hasler, Jun 26 2017

Example for 101053625 corrected by Peter Munn, Jul 23 2017

Status

approved