

A124109


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


1



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
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OFFSET

1,1


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(1p) = p1. 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: (p1)/2 is also prime [then (p1)/2 is called a Sophie Germain prime: see A005384] since as Zak Seidov commented, this is identical to primes p such that p1 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.


LINKS



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.


EXAMPLE

a(1) = 5 because abs(sigma(5)  2*5) = abs(610) = abs(4) = 4 = 2^2 is semiprime.
a(2) = 7 because abs(sigma(7)  2*7) = abs(814) = abs(6) = 6 = 2 * 3 is semiprime.
a(3) = 11 because abs(sigma(11)  2*11) = abs(1222) = abs(10) = 10 = 2 * 5 is semiprime.
a(4) = 12 because abs(sigma(12)  2*12) = abs(2824) = abs(4) = 4 = 2^2 is semiprime.
a(5) = 14 because abs(sigma(14)  2*14) = abs(2428) = abs(+4) = 4 = 2^2 is semiprime.
a(6) = 15 because abs(sigma(15)  2*15) = abs(2430) = abs(6) = 6 = 2 * 3 is semiprime.
a(7) = 21 because abs(sigma(21)  2*21) = abs(3242) = abs(10) = 10 = 2 * 5 is semiprime.
a(8) = 23 because abs(sigma(23)  2*23) = abs(2446) = abs(22) = 22 = 2 * 11 is semiprime.


MATHEMATICA

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


CROSSREFS

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


KEYWORD

easy,nonn


AUTHOR



EXTENSIONS



STATUS

approved



