

A102084


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


7



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,2


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 antiderivative 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 nk and n+k are prime; k = A047160(n). It appears that the antiderivatives of even numbers are overwhelmingly semiprimes of the form n^2  k^2. For example, 1000 has 28 antiderivatives, all of this form. Sequence A189763 lists the even numbers that have antiderivatives not of this form.  T. D. Noe, Apr 27 2011


LINKS

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


FORMULA

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


EXAMPLE

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


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 *)


CROSSREFS

Cf. A073046, A003415, A047160, A099303, A189762.
Sequence in context: A134675 A050530 A278021 * A193315 A030664 A070160
Adjacent sequences: A102081 A102082 A102083 * A102085 A102086 A102087


KEYWORD

nonn


AUTHOR

Michael Taktikos, Feb 16 2005


EXTENSIONS

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


STATUS

approved



