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 A133122 Odd numbers which cannot be written as the sum of an odd prime and 2^i with i > 0. 3
 1, 3, 127, 149, 251, 331, 337, 373, 509, 599, 701, 757, 809, 877, 905, 907, 959, 977, 997, 1019, 1087, 1199, 1207, 1211, 1243, 1259, 1271, 1477, 1529, 1541, 1549, 1589, 1597, 1619, 1649, 1657, 1719, 1759, 1777, 1783, 1807, 1829, 1859, 1867, 1927, 1969, 1973 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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 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. Essentially the same as A006285. - R. J. Mathar, Jun 08 2008 REFERENCES Nathanson, Melvyn B.; Additive Number Theory: The Classical Bases; Springer 1996 Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 62. LINKS J. Z. Schroeder, Every Cubic Bipartite Graph has a Prime Labeling Except K_(3,3), Graphs and Combinatorics (2019) Vol. 35, No. 1, 119-140. EXAMPLE The integer 7 can be represented as 2^2 + 3, therefore it is not on this list. - Michael Taktikos, Feb 02 2009 a(2)=127 because none of the numbers 127-2, 127-4, 127-8, 127-16, 127-32, 127-64 is a prime. MAPLE (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; MATHEMATICA s = {}; Do[Do[s = Union[s, {Prime[n] + 2^i}], {n, 2, 200}], {i, 1, 10}]; Print[Complement[Range[3, 1000, 2], s]] 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 *) CROSSREFS Cf. A006285, A156695. Sequence in context: A213988 A159319 A086154 * A139936 A221637 A142007 Adjacent sequences:  A133119 A133120 A133121 * A133123 A133124 A133125 KEYWORD nonn AUTHOR David S. Newman, Sep 18 2007 EXTENSIONS More terms and corrected definition from Stefan Steinerberger, Sep 24 2007 Edited by N. J. A. Sloane, Feb 12 2009 at the suggestion of R. J. Mathar STATUS approved

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Last modified March 24 12:14 EDT 2019. Contains 321448 sequences. (Running on oeis4.)