%I
%S 2,3,7,11,29,53,107,211,431,853,1709,3433,6857,13709,27427,54851,
%T 109717,219409,438827,877651,1755319,3510623,7021249,14042491,
%U 28084997,56169977,112339957,224679913,449359829,898719707,1797439359,3594878731,7189757483,14379514973,28759029919
%N a(n) is the least prime such that the subsets of { a(1), ..., a(n) } sum up to 2^n different values.
%C Obviously one must exclude previously used primes. Here this is done by requiring 2^n subsets; equivalently, one could require a(n) to be different from preceding terms, or larger than a(n1). (If a value smaller than a(n1) were possible, then a(n1) would not have been the minimal choice.)
%C If we replace "prime" with "noncomposite", the sequence starts 1, 2, 5, 11, 23, 43, 89, 179, 359, 719, 1433, 2879, ... and seems to coincide with A064934, having a different definition, though.
%C The present sequence clearly (cf. a(4)) would not be the same if the definition were changed to "least prime larger than the sum of preceding terms" (as in A064934).
%C It can be seen that a(n) is always very close to Sum_{i=1..n1} a(i). As a consequence, the sequence grows like a(n) ~ 2^(n0.256...) and thus is not a counterexample to Erdős's conjecture mentioned in the cited paper.
%C The sequence of partial sums, s(n) = Sum_{i=1..n} a(i) =
%C (2,5,12,23,52,105,...) is of alternating parity. If s(n)1 is prime, this is an upper bound for a(n+1), since the smallest element of the sequence is 2; e.g., a(4) = s(3)1. Thus if s(n) is even, a(n+1) <= nextprime(s(n)1). If s(n) is odd, then a(n+1) may be nextprime(s(n)+2) (since the value of s(n) itself is never admissible), as in the case of a(3) = 5 + 2 > s(2) = 5, which is prime.
%H S. J. Benkoski and P. Erdős, <a href="http://dx.doi.org/10.1090/S00255718197403477269">On weird and pseudoperfect numbers</a>, Math. Comp., 28 (1974), pp. 617623. <a href="http://www.renyi.hu/~p_erdos/197424.pdf">Alternate link</a>; <a href="http://dx.doi.org/10.1090/S00255718197503604526">1975 corrigendum</a>
%F a(n) > a(n1) and a(n) <= nextprime((Sum_{i=1..n1} a(i))  (1)^n); but in fact a(n) ~ Sum_{i=1..n1} a(i) and thus a(n) ~ constant*2^n.
%e a(1) = 2, the smallest prime, since subsets of {2} are {},{2} summing to 0 resp. 2.
%e a(2) = 3, the second smallest prime, since subsets
%e {},{2},{3},{2,3} have sums 0, 2, 3, 5 which are all different.
%e Then, 5 is not allowed for a(3), since for {2,3,5}, the sum of the subset {2,3} would be the same as that of {5}.
%e For a(3)=7, however, the set of the previously possible sums, {0,2,3,5} and the set of possible sums using the new element, 7 + {0,2,3,5} = {7,9,10,12} are disjoint.
%e Obviously this is always true for a(n) larger than the sum of all preceding terms.
%e However, a(4) = 11 is smaller than this sum (7 + 3 + 2 = 12), yet {0,2,3,5,7,9,10,12} and 11 + {0,2,3,5,7,9,10,12} are disjoint.
%o (PARI) s=p=1; for( n=1,30, while( bitand(s,s>>p=nextprime(p+1)),); s+=s<<p; print1(p","))
%Y Cf. A064934.
%K nonn,changed
%O 1,1
%A _M. F. Hasler_, Apr 08 2008
%E a(23)a(30) from _Donovan Johnson_, Feb 18 2009
%E a(31)a(35) from _Giovanni Resta_, Feb 28 2020
