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

1,2

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

Table of n, a(n) for n=1..47.

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