The OEIS is supported by the many generous donors to the OEIS Foundation.
%I #33 Jun 07 2023 04:37:00
%S 1,2,3,7,13,23,47,97,193,383,769,1531,3067,6133,12269,24533,49069,
%T 98129,196247,392503,785017,1570007,3140041,6280067,12560147,25120289,
%U 50240587,100481167,200962327,401924639,803849303,1607698583,3215397193,6430794373
%N Lexicographically earliest infinite sequence of distinct positive numbers with the property that every positive integer is a sum of distinct terms (see algorithm below).
%C This sequence starts with a(0)=1, subsequent terms a(n) for n > 0 being obtained by selecting the greatest prime <= 1 + Sum_{i=0..n-1} a(i). This ensures that the sequence has the required property because Sum_{i=0..n-1} a(i) >= a(n) - 1, for all n >= 0 and a(0)=1, is a necessary and sufficient condition for it to hold.
%H Charles R Greathouse IV, <a href="/A075058/b075058.txt">Table of n, a(n) for n = 0..1000</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Complete_sequence">"Complete" sequence</a>. [Wikipedia calls a sequence "complete" (sic) if every positive integer is a sum of distinct terms. This name is extremely misleading and should be avoided. - _N. J. A. Sloane_, May 20 2023]
%F a(n) = (greatest prime) <= 1 + Sum_{i=0..n-1} a(i).
%F a(n) ~ k*2^n, with k roughly 0.748643. - _Charles R Greathouse IV_, Apr 05 2013
%e Given that the first 7 terms of the sequence are 1,2,...,23,47 then a(8)=(greatest prime) <= (1+2+...+23,47) + 1 = 97, hence a(8)=97.
%t prevprime[n_Integer] := (j=n; While[!PrimeQ[j], j--]; j) aprime[0]=1; aprime[n_Integer] := (aprime[n] = prevprime[Sum[aprime[m], {m, 0, n - 1}] + 1]); Table[aprime[p], {p, 0, 50}]
%t a[0] = 1; a[n_] := a[n] = NextPrime[Sum[a[k], {k, 0, n-1}]+2, -1]; Table[a[n], {n, 0, 33}] (* _Jean-François Alcover_, Sep 30 2013 *)
%o (PARI) print1(s=1);for(n=1,20,k=precprime(s+1);print1(", "k);s+=k) \\ _Charles R Greathouse IV_, Apr 05 2013
%Y Cf. A068524, A007924, A066352, A200947.
%K nonn
%O 0,2
%A _Amarnath Murthy_, Sep 07 2002
%E Entry revised by _Frank M Jackson_, Dec 03 2011
%E Edited by _N. J. A. Sloane_, May 20 2023