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

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

Offset: 1

Author

David S. Newman, Sep 18 2007

Keywords

• nonn

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

Examples

			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.
		

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.

Crossrefs

Cf. A006285, A156695.

Programs

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 *)

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