cp's OEIS Frontend

a(7) > 10^12. - Donovan Johnson, Feb 29 2012

10^13 < a(7) <= 479535318548480. Other terms of the form 2^k*5*p*q are 983990190817280, 528322308638228480, 1374972658786140160, 9222951307429806080 and 13732480001814200320. - Giovanni Resta, Jul 12 2013

248248256622696037089280 and 29053620223944172891013120 are the next two terms of the form 2^k*5*p*q where p&q are distinct primes. 12, 42 and 1242 are the only terms of one of the three forms 4*p, 2*p*q and 2*p^3*q where p and q are two distinct primes. Farideh Firoozbakht, Aug 19 2013

If p is prime, then the only divisors of p are 1 and p, so sigma(p) = p + 1 and abundance(p) = abs(sigma(p) - 2*p) = abs((p+1) - 2*p) = abs(1-p) = p-1. Hence this sequence includes all values of the sequence of the primes which are one more than semiprimes. This is identical to A005385 Safe primes p: (p-1)/2 is also prime [then (p-1)/2 is called a Sophie Germain prime: see A005384] since as Zak Seidov commented, this is identical to primes p such that p-1 is a semiprime]. But the current sequence also contains composites, such as a(4) = 12, a(5) = 14, a(6) = 15 and a(7) = 21. If k = p*q is a semiprime (with p and q distinct primes) then the only divisors of k are 1, p, q and p*q, so sigma(k) = 1 + p + q + p*q and abs(abundance(k)) = abs(1 + p + q + p*q - p*q) = abs(1 + p + q) and these are in the sequence if 1 + p + q is semiprime. Note that numbers can be in the sequence which are neither prime nor semiprime, starting with a(4) = 12 and a(10) = 27.