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A014092 Numbers that are not the sum of 2 primes.

%I A014092 #57 Jun 08 2022 14:31:14
%S A014092 1,2,3,11,17,23,27,29,35,37,41,47,51,53,57,59,65,67,71,77,79,83,87,89,
%T A014092 93,95,97,101,107,113,117,119,121,123,125,127,131,135,137,143,145,147,
%U A014092 149,155,157,161,163,167,171,173,177,179,185,187,189,191,197,203,205,207,209
%N A014092 Numbers that are not the sum of 2 primes.
%C A014092 Suggested by the Goldbach conjecture that every even number larger than 2 is the sum of 2 primes.
%C A014092 Since (if we believe the Goldbach conjecture) all the entries > 2 in this sequence are odd, they are equal to 2 + an odd composite number (or 1).
%C A014092 Otherwise said, the sequence consists of 2 and odd numbers k such that k-2 is not prime. In particular there is no element from A006512, greater of a twin prime pair. - _M. F. Hasler_, Sep 18 2012
%C A014092 Values of k such that A061358(k) = 0. - _Emeric Deutsch_, Apr 03 2006
%C A014092 Values of k such that A073610(k) = 0. - _Graeme McRae_, Jul 18 2006
%D A014092 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, Section 2.8 (for Goldbach conjecture).
%H A014092 Reinhard Zumkeller, <a href="/A014092/b014092.txt">Table of n, a(n) for n = 1..10000</a>
%H A014092 <a href="/index/Go#Goldbach">Index entries for sequences related to Goldbach conjecture</a>
%F A014092 Odd composite numbers + 2 (essentially A014076(n) + 2 ).
%F A014092 Equals {2} union A005408 \ A052147, i.e., essentially the complement of A052147 (or rather A048974) within the odd numbers A005408. - _M. F. Hasler_, Sep 18 2012
%p A014092 g:=sum(sum(x^(ithprime(i)+ithprime(j)),i=1..j),j=1..50): gser:=series(g,x=0,230): a:=proc(n) if coeff(gser,x^n)=0 then n else fi end: seq(a(n),n=1..225); # _Emeric Deutsch_, Apr 03 2006
%t A014092 s1falsifiziertQ[s_]:= Module[{ip=IntegerPartitions[s, {2}], widerlegt=False},Do[If[PrimeQ[ip[[i,1]] ] ~And~ PrimeQ[ip[[i,2]] ], widerlegt = True; Break[]],{i,1,Length[ip]}];widerlegt]; Select[Range[250],s1falsifiziertQ[ # ]==False&] (* _Michael Taktikos_, Dec 30 2007 *)
%t A014092 Join[{1,2},Select[Range[3,300,2],!PrimeQ[#-2]&]] (* _Zak Seidov_, Nov 27 2010 *)
%t A014092 Select[Range[250],Count[IntegerPartitions[#,{2}],_?(AllTrue[#,PrimeQ]&)]==0&] (* _Harvey P. Dale_, Jun 08 2022 *)
%o A014092 (PARI) isA014092(n)=local(p,i) ; i=1 ; p=prime(i); while(p<n, if( isprime(n-p), return(0)); i++; p=prime(i)); 1
%o A014092 n=1; for(a=1,200, if(isA014092(a), print(n," ",a); n++)) \\ _R. J. Mathar_, Aug 20 2006
%o A014092 (Haskell)
%o A014092 a014092 n = a014092_list !! (n-1)
%o A014092 a014092_list = filter (\x ->
%o A014092    all ((== 0) . a010051) $ map (x -) $ takeWhile (< x) a000040_list) [1..]
%o A014092 -- _Reinhard Zumkeller_, Sep 28 2011
%o A014092 (Python)
%o A014092 from sympy import prime, isprime
%o A014092 def ok(n):
%o A014092     i=1
%o A014092     x=prime(i)
%o A014092     while x<n:
%o A014092         if isprime(n - x): return False
%o A014092         i+=1
%o A014092         x=prime(i)
%o A014092     return True
%o A014092 print([n for n in range(1, 301) if ok(n)]) # _Indranil Ghosh_, Apr 29 2017
%Y A014092 Cf. A002372, A002373, A002374, A048974, A061358.
%Y A014092 Cf. A010051, A000040, A051035 (composites).
%Y A014092 Equivalent sequence for prime powers: A071331.
%Y A014092 Numbers that can be expressed as the sum of two primes in k ways for k=0..10: this sequence (k=0), A067187 (k=1), A067188 (k=2), A067189 (k=3), A067190 (k=4), A067191 (k=5), A066722 (k=6), A352229 (k=7), A352230 (k=8), A352231 (k=9), A352233 (k=10).
%K A014092 nonn,nice,easy
%O A014092 1,2
%A A014092 _N. J. A. Sloane_