A014092 Numbers that are not the sum of 2 primes.

1, 2, 3, 11, 17, 23, 27, 29, 35, 37, 41, 47, 51, 53, 57, 59, 65, 67, 71, 77, 79, 83, 87, 89, 93, 95, 97, 101, 107, 113, 117, 119, 121, 123, 125, 127, 131, 135, 137, 143, 145, 147, 149, 155, 157, 161, 163, 167, 171, 173, 177, 179, 185, 187, 189, 191, 197, 203, 205, 207, 209

Offset

1,2

Comments

Suggested by the Goldbach conjecture that every even number larger than 2 is the sum of 2 primes.

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).

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

Values of k such that A061358(k) = 0. - Emeric Deutsch, Apr 03 2006

Values of k such that A073610(k) = 0. - Graeme McRae, Jul 18 2006

References

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).

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Index entries for sequences related to Goldbach conjecture

Formula

Odd composite numbers + 2 (essentially A014076(n) + 2 ).

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

Maple

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

Mathematica

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 *)
Join[{1, 2}, Select[Range[3, 300, 2], !PrimeQ[#-2]&]] (* Zak Seidov, Nov 27 2010 *)
Select[Range[250], Count[IntegerPartitions[#, {2}], _?(AllTrue[#, PrimeQ]&)]==0&] (* Harvey P. Dale, Jun 08 2022 *)

Prog

(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
n=1; for(a=1, 200, if(isA014092(a), print(n, " ", a); n++)) \\ R. J. Mathar, Aug 20 2006
(Haskell)
a014092 n = a014092_list !! (n-1)
a014092_list = filter (\x ->
   all ((== 0) . a010051) $ map (x -) $ takeWhile (< x) a000040_list) [1..]
-- Reinhard Zumkeller, Sep 28 2011
(Python)
from sympy import prime, isprime
def ok(n):
    i=1
    x=prime(i)
    while x<n:
        if isprime(n - x): return False
        i+=1
        x=prime(i)
    return True
print([n for n in range(1, 301) if ok(n)]) # Indranil Ghosh, Apr 29 2017

Crossrefs

Cf. A002372, A002373, A002374, A048974, A061358.

Cf. A010051, A000040, A051035 (composites).

Equivalent sequence for prime powers: A071331.

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).

Sequence in context: A215811 A051076 A192612 * A100962 A045337 A098700

Adjacent sequences: A014089 A014090 A014091 * A014093 A014094 A014095

Keyword

nonn,nice,easy

Author

N. J. A. Sloane

Status

approved