This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A025583 #53 Feb 16 2025 08:32:35
%S A025583 27,35,51,57,65,77,87,93,95,117,119,121,123,125,135,143,145,147,155,
%T A025583 161,171,177,185,187,189,203,205,207,209,215,217,219,221,237,245,247,
%U A025583 249,255,261,267,275,287,289,291,297,299,301,303,305,321,323,325,327,329,335,341
%N A025583 Composite numbers that are not the sum of 2 primes.
%C A025583 Goldbach conjectured that every integer > 5 is the sum of three primes.
%C A025583 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 A025583 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 A025583 T. D. Noe, <a href="/A025583/b025583.txt">Table of n, a(n) for n = 1..10000</a>
%H A025583 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimePartition.html">Prime Partition</a>
%H A025583 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TwinComposites.html">Twin Composites</a>
%H A025583 <a href="/index/Go#Goldbach">Index entries for sequences related to Goldbach conjecture</a>
%t A025583 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 A025583 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 A025583 Select[Range[400],CompositeQ[#]&&Count[IntegerPartitions[#,{2}],_?(AllTrue[ #,PrimeQ]&)]==0&] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Feb 21 2021 *)
%o A025583 (Haskell)
%o A025583 a025583 n = a025583_list !! (n-1)
%o A025583 a025583_list = filter f a002808_list where
%o A025583 f x = all (== 0) $ map (a010051 . (x -)) $ takeWhile (< x) a000040_list
%o A025583 -- _Reinhard Zumkeller_, Oct 15 2014
%Y A025583 Cf. A051034, A001031, A002372, A002374, A071335.
%Y A025583 Cf. A002808, A000040, A010051.
%K A025583 nonn,nice,easy
%O A025583 1,1
%A A025583 _N. J. A. Sloane_