A080192 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.

59, 71, 101, 107, 149, 263, 311, 347, 461, 499, 521, 569, 673, 757, 821, 823, 857, 881, 883, 907, 967, 977, 1009, 1061, 1091, 1093, 1151, 1213, 1279, 1283, 1297, 1301, 1319, 1433, 1487, 1489, 1493, 1549, 1571, 1597, 1619, 1667, 1697, 1721, 1787, 1871, 1873

Offset

1,1

Comments

From Peter Munn, Oct 19 2017: (Start)

This is also a list of the leaf node labels in the tree of primes described in A290183.

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.

(End)

From Peter Munn, Nov 02 2017: (Start)

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).

If p is the lesser of twin primes it is in this sequence if it is neither Sophie Germain nor in A023204.

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.

(End)

Links

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from Michel Marcus)

Formula

For all k, prime(k) = A000040(k) is a term if and only if A102820(k) = 0. - Peter Munn, Oct 24 2017

Example

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.

Mathematica

Select[Prime[Range[300]], NextPrime[2#]>2NextPrime[#]&] (* Harvey P. Dale, Jul 07 2011 *)

Prog

(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, ", ")))}
(PARI) isok(p) = isprime(p) && (primepi(2*p) == primepi(2*nextprime(p+1)));
forprime(p=2, 2000, if (isok(p), print1(p, ", "))) \\ Michel Marcus, Sep 22 2017
(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
(NARS2000) ¯1↓b/⍨(1⌽a)<1πa←2×b←¯2π⍳1E4 ⍝ Michael Turniansky, Dec 29 2020

Crossrefs

A080191 is the complement of this sequence relative to A000040.

Sequences with related analysis: A005384, A023204, A052248, A102820, A215238, A290183.

Sequences with similar definitions: A195270, A195271, A195325, A195377.

Sequence in context: A061764 A316971 A162000 * A126693 A139938 A033235

Adjacent sequences: A080189 A080190 A080191 * A080193 A080194 A080195

Keyword

nonn

Author

Klaus Brockhaus, Feb 10 2003

Status

approved