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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..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.
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 0..1000
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..n-1} a(i).
a(n) ~ k*2^n, with k roughly 0.748643. - Charles R Greathouse IV, Apr 05 2013
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, n-1}]+2, -1]; Table[a[n], {n, 0, 33}] (* Jean-François Alcover, Sep 30 2013 *)
PROG
(PARI) print1(s=1); for(n=1, 20, k=precprime(s+1); print1(", "k); s+=k) \\ Charles R Greathouse IV, Apr 05 2013
CROSSREFS
Sequence in context: A091440 A175211 A330028 * A213968 A213967 A128695
KEYWORD
nonn
AUTHOR
Amarnath Murthy, Sep 07 2002
EXTENSIONS
Entry revised by Frank M Jackson, Dec 03 2011
Edited by N. J. A. Sloane, May 20 2023
STATUS
approved

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Last modified July 13 14:24 EDT 2024. Contains 374284 sequences. (Running on oeis4.)