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%I #23 Jul 30 2015 21:49:42
%S 0,4,9,15,25,35,49,55,77,91,121,143,169,187,221,247,289,323,361,391,
%T 437,403,529,551,589,667,713,703,841,899,961,943,1073,1147,1189,1271,
%U 1369,1363,1517,1591,1681,1763,1849,1927,2021,1891,2209,2279,2257,2491
%N a(1) = 0; for n>0, write 2n=p+q (p, q prime), p*q maximal; then a(n)=p*q (see A073046).
%C For n>1, largest semiprime whose sum of prime factors = 2n. Assumes the Goldbach conjecture is true. Also the largest semiprime <= n^2.
%C 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
%H T. D. Noe, <a href="/A102084/b102084.txt">Table of n, a(n) for n = 1..1000</a>
%F a(n) = n^2 - A047160(n)^2. - _Jason Kimberley_, Jun 26 2012
%e n=13: 2n = 26; 26 = 23 + 3 = 19 + 7 = 13 + 13; 13*13 = maximal => p*q = 13*13 = 169.
%t 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 *)
%t 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 *)
%Y Cf. A073046, A003415, A047160, A099303, A189762.
%K nonn
%O 1,2
%A _Michael Taktikos_, Feb 16 2005
%E Edited by _N. J. A. Sloane_, Aug 29 2008 at the suggestion of _R. J. Mathar_
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