A124109 - OEIS

%I #11 Aug 03 2019 07:57:48

%S 5,7,11,12,14,15,21,23,26,27,34,35,39,40,44,47,52,55,57,58,59,63,65,

%T 68,70,72,74,75,77,80,82,83,85,88,93,98,107,110,115,116,119,122,125,

%U 129,133,143,144,152,155,160,162,164,167,169,171,178,179,183,185,187,189

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

%C 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.

%H Amiram Eldar, <a href="/A124109/b124109.txt">Table of n, a(n) for n = 1..10000</a>

%F 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.

%e a(1) = 5 because abs(sigma(5) - 2*5) = abs(6-10) = abs(-4) = 4 = 2^2 is semiprime.

%e a(2) = 7 because abs(sigma(7) - 2*7) = abs(8-14) = abs(-6) = 6 = 2 * 3 is semiprime.

%e a(3) = 11 because abs(sigma(11) - 2*11) = abs(12-22) = abs(-10) = 10 = 2 * 5 is semiprime.

%e a(4) = 12 because abs(sigma(12) - 2*12) = abs(28-24) = abs(-4) = 4 = 2^2 is semiprime.

%e a(5) = 14 because abs(sigma(14) - 2*14) = abs(24-28) = abs(+4) = 4 = 2^2 is semiprime.

%e a(6) = 15 because abs(sigma(15) - 2*15) = abs(24-30) = abs(-6) = 6 = 2 * 3 is semiprime.

%e a(7) = 21 because abs(sigma(21) - 2*21) = abs(32-42) = abs(-10) = 10 = 2 * 5 is semiprime.

%e a(8) = 23 because abs(sigma(23) - 2*23) = abs(24-46) = abs(-22) = 22 = 2 * 11 is semiprime.

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

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

%K easy,nonn

%O 1,1

%A _Jonathan Vos Post_, Nov 26 2006

%E More terms from _Robert G. Wilson v_, Nov 29 2006