%I A138000
%S A138000 2,3,7,11,29,53,107,211,431,853,1709,3433,6857,13709,27427,54851,109717,
%T A138000 219409,438827,877651,1755319,3510623,7021249,14042491,28084997,
%U A138000 56169977,112339957,224679913,449359829,898719707
%N A138000 Least prime such that the subsets of { a(1),...,a(n) } sum up to 2^n 
               different values.
%C A138000 Obviously one must exclude previously used primes. Here
%C A138000 this is done by requiring 2^n
%C A138000 subsets, equivalently one could require a(n) to be different from
%C A138000 preceding terms, or larger than a(n-1).
%C A138000 (If a value smaller than a(n-1) were possible, then a(n-1) would not
%C A138000 have been the minimal choice.)
%C A138000 If we replace "prime" by "non-composite", the sequence
%C A138000 starts 1, 2, 5, 11, 23, 43, 89, 179, 359, 719, 1433, 2879,... and
%C A138000 seems to coincide with A064934, having a different definition, though.
%C A138000 The present sequence clearly (cf. a(4)) would not be the same if the
%C A138000 definition be changed to "least prime larger than the sum of preceding
%C A138000 terms" (as in A064934).
%C A138000 It can be seen that a(n) is always very close to sum(a(i),i=1..n-1).
%C A138000 As a consequence, the sequence grows like a(n) ~ 2^(n-0.256...) and
%C A138000 thus is not a counter-example to Erdos' conjecture mentioned in the
%C A138000 cited paper.
%C A138000 The sequence of partial sums, s(n) = sum(a(i),i=1..n) =
%C A138000 (2,5,12,23,52,105,...) is of alternating parity. If s(n)-1 is prime,
%C A138000 this is an upper bound for a(n+1), since the smallest element of the
%C A138000 sequence is 2; e.g., a(4)=s(3)-1. Thus if s(n) is even, a(n+1)<=nextprime(s(n)-1).
%C A138000 If s(n) is odd, then a(n+1) may be nextprime(s(n)+2) (since the value
%C A138000 of s(n) itself is never admissible),
%C A138000 as in the case of a(3)=5+2>s(2)=5 which is prime.
%D A138000 S. J. Benkoski and P. Erdos, On weird and pseudoperfect numbers, Math. 
               Comp. 28 (1974) 617-623.
%F A138000 a(n) > a(n-1) and a(n) <= nextprime(sum(a(i),i=1..n-1)-(-1)^n) ; but 
               in fact a(n) ~ sum(a(i),i=1..n-1) and thus a(n) ~ constant*2^n
%e A138000 a(1)=2, the smallest prime, since subsets of {2} are {},{2}
%e A138000 summing to 0 resp. 2.
%e A138000 a(2)=3, the second smallest prime, since subsets
%e A138000 {},{2},{3},{2,3} have sums 0, 2, 3, 5 which are all different.
%e A138000 Then, 5 is not allowed for a(3), since for {2,3,5}, the sum
%e A138000 of the subset {2,3} would be the same than that of {5}.
%e A138000 For a(3)=7, however, the set of the previously possible sums,
%e A138000 {0,2,3,5} and the set of possible sums using the new element,
%e A138000 7+{0,2,3,5} = {7,9,10,12} are disjoint.
%e A138000 Obviously this is always true for a(n) larger than the sum of all
%e A138000 preceding terms.
%e A138000 However, a(4)=11 is smaller than this sum (7+3+2=12);
%e A138000 but yet {0,2,3,5,7,9,10,12} and 11+{0,2,3,5,7,9,10,12} are disjoint.
%o A138000 (PARI) s=p=1; for( n=1,30, while( bitand(s,s>>p=nextprime(p+1)),); s+=s<<p; 
               print1(p","))
%Y A138000 Cf. A064934.
%Y A138000 Sequence in context: A096362 A005479 A120856 this_sequence A034295 A140108 
               A056354
%Y A138000 Adjacent sequences: A137997 A137998 A137999 this_sequence A138001 A138002 
               A138003
%K A138000 nonn,more
%O A138000 1,1
%A A138000 M. F. Hasler (www.univ-ag.fr/~mhasler), Apr 08 2008
%E A138000 a(23)-a(30) from Donovan Johnson (donovan.johnson(AT)yahoo.com), Feb 
               18 2009