A201828 The smallest A(m) such that the interval (A(m)*n, A(m+1)*n) contains exactly one element of A, where A is the sequence of primes p for which p-2 is not prime.

37, 37, 2, 2, 2, 2, 907, 2, 2833, 907, 2, 8269, 2749, 2953, 5413, 7699, 2137, 27103, 28513, 74377, 45673, 56629, 79147, 33529, 15259, 96847, 101599, 57649, 44983, 300973, 706309, 715357, 351847, 38557, 308809, 720607, 901447, 2229889, 867253, 2370937, 1276867

Offset

2,1

Comments

This sequence is the "A-analog" of A195871.

This is a possible model sequence to understand the role of twin primes in sequences like A195871. In particular, if after a large number N_tw, there are no twin primes, what primes will take their place in A195871? Our observations and expectations are expressed in the following conjecture.

Conjecture: For n>=13, every a(n) is the lesser of a pair of cousin primes p and p+4, cf. A023200. Note that it is only conjectured that there are infinitely many pairs of cousin primes.

The limit of a(n) as n goes to infinity is infinity.

Links

Table of n, a(n) for n=2..42.

Example

Let n=2. We have the following intervals of the form (2*p,2*q), where p,q are consecutive primes in A025584:(4,6),(6,22),(22,34),(34,46),(46,58),(58,74),(74,82),..., containing 0,2,2,2,2,3,1,... primes from A025584. The interval (74,82) is the first to contain exactly one prime from A025584, so a(2)=74/2=37.

Mathematica

myPrime=Select[#, !PrimeQ[#-2]&]&[Prime[Range[500000]]];  binarySearch[lst_, find_]:=Module[{lo=1, up=Length[lst], v}, (While[lo<=up, v=Floor[(lo+up)/2]; If[lst[[v]]-find==0, Return[v]]; If[lst[[v]]<find, lo=v+1, up=v-1]]; 0)]; myPrimeQ[n_]:=binarySearch[myPrime, n]; nextMyPrime[n_, offset_Integer:1]:=myPrime[[myPrimeQ[NextPrime[n, NestWhile[#1+1&, 1, !myPrimeQ[NextPrime[n, #1]]>0&]]]+offset-1]];   z=1; (*example for "contains exactly ONE myPrime in the interval"*)Table[myPrime[[NestWhile[#1+1&, 1, !((nextMyPrime[n myPrime[[#1]], z]<n myPrime[[#1+1]]&&nextMyPrime[n myPrime[[#1]], z+1]>n myPrime[[#1+1]]))&]]], {n, 2, 30}]

Prog

(PARI) npr(n) = {local(p); p=n+1; while(!isprime(p) || isprime(p-2), p=p+1); p}
cnt(a, b) = {local(r); r=0; for(p=a, b, if(isprime(p) && !isprime(p-2), r=r+1)); r}
a201828(n) = {local(a, b); a=2; b=3; while(cnt(a*n, b*n) != 1, a=b; b=npr(b)); a} \\ Michael B. Porter, Feb 18 2013

Crossrefs

Cf. A025584, A195871, A023200.

Sequence in context: A324249 A022993 A023479 * A291478 A257810 A090179

Adjacent sequences: A201825 A201826 A201827 * A201829 A201830 A201831

Keyword

nonn

Author

Vladimir Shevelev and Peter J. C. Moses, Jan 09 2013

Status

approved