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A133122 Odd numbers which cannot be written as the sum of an odd prime and 2^i with i > 0.

%I A133122 #22 Jan 23 2019 03:06:16
%S A133122 1,3,127,149,251,331,337,373,509,599,701,757,809,877,905,907,959,977,
%T A133122 997,1019,1087,1199,1207,1211,1243,1259,1271,1477,1529,1541,1549,1589,
%U A133122 1597,1619,1649,1657,1719,1759,1777,1783,1807,1829,1859,1867,1927,1969,1973
%N A133122 Odd numbers which cannot be written as the sum of an odd prime and 2^i with i > 0.
%C A133122 The sequence of "obstinate numbers", that is, odd numbers which cannot be written as prime + 2^i with i >= 0 is the same except for the initial 3. - _N. J. A. Sloane_, Apr 20 2008
%C A133122 The reference by Nathanson gives on page 206 a theorem of Erdos: There exists an infinite arithmetic progression of odd positive integers, none of which is of the form p+2^k.
%C A133122 Essentially the same as A006285. - _R. J. Mathar_, Jun 08 2008
%D A133122 Nathanson, Melvyn B.; Additive Number Theory: The Classical Bases; Springer 1996
%D A133122 Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 62.
%H A133122 J. Z. Schroeder, <a href="https://doi.org/10.1007/s00373-018-1980-y">Every Cubic Bipartite Graph has a Prime Labeling Except K_(3,3)</a>, Graphs and Combinatorics (2019) Vol. 35, No. 1, 119-140.
%e A133122 The integer 7 can be represented as 2^2 + 3, therefore it is not on this list. - _Michael Taktikos_, Feb 02 2009
%e A133122 a(2)=127 because none of the numbers 127-2, 127-4, 127-8, 127-16, 127-32, 127-64 is a prime.
%p A133122 (Maple program which returns -1 iff 2n+1 is obstinate, from _N. J. A. Sloane_, Apr 20 2008): f:=proc(n) local i,t1; t1:=2*n+1; i:=0; while 2^i < t1 do if isprime(t1-2^i) then RETURN(1); fi; i:=i+1; end do; RETURN(-1); end proc;
%t A133122 s = {}; Do[Do[s = Union[s, {Prime[n] + 2^i}], {n, 2, 200}], {i, 1, 10}]; Print[Complement[Range[3, 1000, 2], s]]
%t A133122 zweier = Map[2^# &, Range[0,30]]; primes = Table[Prime[i], {i, 1, 300}]; summen = Union[Flatten[ Table[zweier[[i]] + primes[[j]], {i, 1, 30}, {j, 1, 300}]]]; us = Select[summen, OddQ[ # ] &]; odds = Range[1, 1001, 2]; Complement[odds, us] (* _Michael Taktikos_, Feb 02 2009 *)
%Y A133122 Cf. A006285, A156695.
%K A133122 nonn
%O A133122 1,2
%A A133122 _David S. Newman_, Sep 18 2007
%E A133122 More terms and corrected definition from _Stefan Steinerberger_, Sep 24 2007
%E A133122 Edited by _N. J. A. Sloane_, Feb 12 2009 at the suggestion of _R. J. Mathar_