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
FORMULA
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
KEYWORD
nonn,nice,easy
AUTHOR
STATUS
approved