OFFSET
1,1
COMMENTS
Goldbach conjectured that every integer > 5 is the sum of three primes.
Conjecture: This is the sequence of odd numbers k such that (k mod x) mod 2 != 1, where x is the greatest m <= k such that m, m-1 and m-2 are all composite. Verified for first 10000 terms. - Benedict W. J. Irwin, May 06 2016
Numbers k, such that however many of k coins are placed with heads rather than tails showing, either those showing heads or those showing tails can be arranged in a rectangular pattern with multiple rows and columns. (If the Goldbach conjecture for even numbers is false this comment should be restricted to the odd terms of this sequence, as it might otherwise define a variant sequence). - Peter Munn, May 15 2017
LINKS
T. D. Noe, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime Partition
Eric Weisstein's World of Mathematics, Twin Composites
MATHEMATICA
f[n_] := (p = 0; pn = PrimePi[n]; Do[ If[n == Prime[i] + Prime[k], p = p + 1; If[p > 2, Break[]]], {i, 1, pn}, {k, i, pn}]; p ); Select[Range[2, 400], ! PrimeQ[#] && f[#] == 0 & ] (* Jean-François Alcover, Mar 07 2011 *)
upto=350; With[{c=PrimePi[upto]}, Complement[Range[4, upto], Prime[Range[ c]], Union[Total/@Tuples[Prime[Range[c]], {2}]]]] (* Harvey P. Dale, Jul 14 2011 *)
Select[Range[400], CompositeQ[#]&&Count[IntegerPartitions[#, {2}], _?(AllTrue[ #, PrimeQ]&)]==0&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Feb 21 2021 *)
PROG
(Haskell)
a025583 n = a025583_list !! (n-1)
a025583_list = filter f a002808_list where
f x = all (== 0) $ map (a010051 . (x -)) $ takeWhile (< x) a000040_list
-- Reinhard Zumkeller, Oct 15 2014
CROSSREFS
KEYWORD
nonn,nice,easy
AUTHOR
STATUS
approved