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%I #37 Apr 24 2023 12:13:00
%S 59,71,101,107,149,263,311,347,461,499,521,569,673,757,821,823,857,
%T 881,883,907,967,977,1009,1061,1091,1093,1151,1213,1279,1283,1297,
%U 1301,1319,1433,1487,1489,1493,1549,1571,1597,1619,1667,1697,1721,1787,1871,1873
%N Complement of A080191 relative to A000040. Prime p is a term iff there is no prime between 2*p and 2*q, where q is the next prime after p.
%C From _Peter Munn_, Oct 19 2017: (Start)
%C This is also a list of the leaf node labels in the tree of primes described in A290183.
%C For k > 0, the earliest run of k adjacent primes in this sequence starts with the least prime greater than A215238(k+1)/2. Thus we see that A215238(3) = 1637 corresponds to 821 followed by 823 being the first run of 2 adjacent primes in this sequence.
%C (End)
%C From _Peter Munn_, Nov 02 2017: (Start)
%C If p is in A005384 (a Sophie Germain prime), 2p+1 is therefore a prime, so p cannot be in this sequence. Similarly, any prime p in A023204 has a corresponding prime 2p+3, which (if p>2) likewise implies its absence (and if p=2 it is in A005384).
%C If p is the lesser of twin primes it is in this sequence if it is neither Sophie Germain nor in A023204.
%C Conjecture: a(n)/A000040(n) is asymptotic to 3. Reason: I expect the distribution of terms in A102820 to converge to a geometric distribution with mean value 2.
%C (End)
%H David A. Corneth, <a href="/A080192/b080192.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from Michel Marcus)
%F For all k, prime(k) = A000040(k) is a term if and only if A102820(k) = 0. - _Peter Munn_, Oct 24 2017
%e 59 is a term since 113 is the prime preceding 2*59, 127 is the next prime and 61 is the largest of all prime factors of 114, ..., 122 = 2*61, ..., 126.
%t Select[Prime[Range[300]],NextPrime[2#]>2NextPrime[#]&] (* _Harvey P. Dale_, Jul 07 2011 *)
%o (PARI) {forprime(k=2,1873,p=precprime(2*k); q=nextprime(p+1); m=0; for(j=p+1,q-1,f=factor(j); a=f[matsize(f)[1],1]; if(m<a,m=a)); if(m!=k,print1(k,",")))}
%o (PARI) isok(p) = isprime(p) && (primepi(2*p) == primepi(2*nextprime(p+1)));
%o forprime(p=2, 2000, if (isok(p), print1(p, ", "))) \\ _Michel Marcus_, Sep 22 2017
%o (PARI) first(n) = my(res = vector(n), i = 0); {n==0&&return([]); forprime(p = 2, , if(nextprime(2*p) > 2*nextprime(p + 1), i++; res[i] = p; if(i == n, return(res))))} \\ _David A. Corneth_, Oct 25 2017
%o (NARS2000) ¯1↓b/⍨(1⌽a)<1πa←2×b←¯2π⍳1E4 ⍝ _Michael Turniansky_, Dec 29 2020
%Y A080191 is the complement of this sequence relative to A000040.
%Y Sequences with related analysis: A005384, A023204, A052248, A102820, A215238, A290183.
%Y Sequences with similar definitions: A195270, A195271, A195325, A195377.
%K nonn
%O 1,1
%A _Klaus Brockhaus_, Feb 10 2003
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