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A075058 1 followed by the greatest primes selected to form a complete sequence (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 at a(0)=1, subsequent terms a(n) for n>0 being obtained by selecting the (greatest prime) <= Sum(a(i),(i,0,n-1))+1. This ensures that the sequence is complete because Sum(a(i),(i,0,n-1))>=a(n)-1, for all n>=0 and a(0)=1, is a necessary and sufficient condition for completeness.

LINKS

Charles R Greathouse IV, Table of n, a(n) for n = 0..1000

Wikipedia, Complete sequence

FORMULA

a(n) = (greatest prime) <= Sum(a(i),(i,0,n-1))+1.

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-Fran├žois 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

Cf. A068524, A007924, A066352, A200947.

Sequence in context: A091440 A175211 A330028 * A213968 A213967 A128695

Adjacent sequences:  A075055 A075056 A075057 * A075059 A075060 A075061

KEYWORD

nonn

AUTHOR

Amarnath Murthy, Sep 07 2002

EXTENSIONS

Entry revised by Frank M Jackson, Dec 03 2011

STATUS

approved

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Last modified September 20 13:19 EDT 2020. Contains 337264 sequences. (Running on oeis4.)