A124109 Numbers whose abundance (A033880) or deficiency (A033879) is a semiprime number.

5, 7, 11, 12, 14, 15, 21, 23, 26, 27, 34, 35, 39, 40, 44, 47, 52, 55, 57, 58, 59, 63, 65, 68, 70, 72, 74, 75, 77, 80, 82, 83, 85, 88, 93, 98, 107, 110, 115, 116, 119, 122, 125, 129, 133, 143, 144, 152, 155, 160, 162, 164, 167, 169, 171, 178, 179, 183, 185, 187, 189

Offset: 1

Author

Jonathan Vos Post, Nov 26 2006

Keywords

• easy
• nonn

Comments

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.

Examples

			a(1) = 5 because abs(sigma(5) - 2*5) = abs(6-10) = abs(-4) = 4 = 2^2 is semiprime.
a(2) = 7 because abs(sigma(7) - 2*7) = abs(8-14) = abs(-6) = 6 = 2 * 3 is semiprime.
a(3) = 11 because abs(sigma(11) - 2*11) = abs(12-22) = abs(-10) = 10 = 2 * 5 is semiprime.
a(4) = 12 because abs(sigma(12) - 2*12) = abs(28-24) = abs(-4) = 4 = 2^2 is semiprime.
a(5) = 14 because abs(sigma(14) - 2*14) = abs(24-28) = abs(+4) = 4 = 2^2 is semiprime.
a(6) = 15 because abs(sigma(15) - 2*15) = abs(24-30) = abs(-6) = 6 = 2 * 3 is semiprime.
a(7) = 21 because abs(sigma(21) - 2*21) = abs(32-42) = abs(-10) = 10 = 2 * 5 is semiprime.
a(8) = 23 because abs(sigma(23) - 2*23) = abs(24-46) = abs(-22) = 22 = 2 * 11 is semiprime.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000203, A001358, A000040, A005384, A005385, A077374, A087998, A088005, A088006, A125236, A125237.

Programs

Mathematica

semiPrimeQ[n_] := Plus @@ Last /@ FactorInteger@n == 2; Select[ Range@ 193, semiPrimeQ@Abs[DivisorSigma[1, # ] - 2# ] &] (* Robert G. Wilson v *)

Formula

Abs[sigma(a(n)) - 2*a(n)] is a semiprime, where sigma(k) = sum of divisors of k. {Abs[sigma(a(n)) - 2*a(n)]} is in A001358.

Extensions

More terms from Robert G. Wilson v, Nov 29 2006