A102084 a(1) = 0; for n>0, write 2n=p+q (p, q prime), p*q maximal; then a(n)=p*q (see A073046).

0, 4, 9, 15, 25, 35, 49, 55, 77, 91, 121, 143, 169, 187, 221, 247, 289, 323, 361, 391, 437, 403, 529, 551, 589, 667, 713, 703, 841, 899, 961, 943, 1073, 1147, 1189, 1271, 1369, 1363, 1517, 1591, 1681, 1763, 1849, 1927, 2021, 1891, 2209, 2279, 2257, 2491

Offset: 1

Author

Michael Taktikos, Feb 16 2005

Keywords

• nonn

Comments

For n>1, largest semiprime whose sum of prime factors = 2n. Assumes the Goldbach conjecture is true. Also the largest semiprime <= n^2.

Also the greatest integer x such that x' = 2*n, or 0 if there is no such x, where x' is the arithmetic derivative (A003415). Bisection of A099303. The only even number without an anti-derivative is 2. All terms are <= n^2, with equality only when n is prime. In fact a(n) = n^2 - k^2, where k is the least number such that both n-k and n+k are prime; k = A047160(n). It appears that the anti-derivatives of even numbers are overwhelmingly semiprimes of the form n^2 - k^2. For example, 1000 has 28 anti-derivatives, all of this form. Sequence A189763 lists the even numbers that have anti-derivatives not of this form. - T. D. Noe, Apr 27 2011

Examples

			n=13: 2n = 26; 26 = 23 + 3 = 19 + 7 = 13 + 13; 13*13 = maximal => p*q = 13*13 = 169.
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A073046, A003415, A047160, A099303, A189762.

Programs

Mathematica

f[n_] := Block[{pf = FactorInteger[n]}, If[Plus @@ Last /@ pf == 2, If[ Length[pf] == 2, Plus @@ First /@ pf, 2pf[[1, 1]]], 0]]; t = Table[0, {51}]; Do[a = f[n]; If[ EvenQ[a] && 0 < a < 104, t[[a/2]] = n], {n, 2540}]; t (* Robert G. Wilson v, Jun 14 2005 *)
Table[k = 0; While[k < n && (! PrimeQ[n - k] || ! PrimeQ[n + k]), k++]; If[k == n, 0, (n - k)*(n + k)], {n, 100}] (* T. D. Noe, Apr 27 2011 *)

Formula

a(n) = n^2 - A047160(n)^2. - Jason Kimberley, Jun 26 2012

Extensions

Edited by N. J. A. Sloane, Aug 29 2008 at the suggestion of R. J. Mathar