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%I #52 Dec 17 2021 10:53:28
%S 27,35,51,57,65,77,87,93,95,117,119,121,123,125,135,143,145,147,155,
%T 161,171,177,185,187,189,203,205,207,209,215,217,219,221,237,245,247,
%U 249,255,261,267,275,287,289,291,297,299,301,303,305,321,323,325,327,329,335,341
%N Composite numbers that are not the sum of 2 primes.
%C Goldbach conjectured that every integer > 5 is the sum of three primes.
%C 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
%C 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
%H T. D. Noe, <a href="/A025583/b025583.txt">Table of n, a(n) for n = 1..10000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimePartition.html">Prime Partition</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TwinComposites.html">Twin Composites</a>
%H <a href="/index/Go#Goldbach">Index entries for sequences related to Goldbach conjecture</a>
%t 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 *)
%t 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 *)
%t Select[Range[400],CompositeQ[#]&&Count[IntegerPartitions[#,{2}],_?(AllTrue[ #,PrimeQ]&)]==0&] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Feb 21 2021 *)
%o (Haskell)
%o a025583 n = a025583_list !! (n-1)
%o a025583_list = filter f a002808_list where
%o f x = all (== 0) $ map (a010051 . (x -)) $ takeWhile (< x) a000040_list
%o -- _Reinhard Zumkeller_, Oct 15 2014
%Y Cf. A051034, A001031, A002372, A002374, A071335.
%Y Cf. A002808, A000040, A010051.
%K nonn,nice,easy
%O 1,1
%A _N. J. A. Sloane_
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