

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
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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.


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



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 1272, 1274, 1278, 12716, 12732, 12764 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(t12^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



KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



