

A075058


Lexicographically earliest infinite sequence of distinct positive numbers with the property that every positive integer is a sum of distinct terms (see algorithm below).


7



1, 2, 3, 7, 13, 23, 47, 97, 193, 383, 769, 1531, 3067, 6133, 12269, 24533, 49069, 98129, 196247, 392503, 785017, 1570007, 3140041, 6280067, 12560147, 25120289, 50240587, 100481167, 200962327, 401924639, 803849303, 1607698583, 3215397193, 6430794373
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,2


COMMENTS

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..n1} a(i). This ensures that the sequence has the required property because Sum_{i=0..n1} a(i) >= a(n)  1, for all n >= 0 and a(0)=1, is a necessary and sufficient condition for it to hold.


LINKS

Wikipedia, "Complete" sequence. [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]


FORMULA

a(n) = (greatest prime) <= 1 + Sum_{i=0..n1} a(i).


EXAMPLE

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.


MATHEMATICA

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}]
a[0] = 1; a[n_] := a[n] = NextPrime[Sum[a[k], {k, 0, n1}]+2, 1]; Table[a[n], {n, 0, 33}] (* JeanFrançois Alcover, Sep 30 2013 *)


PROG



CROSSREFS



KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



