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A201828
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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.
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1
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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
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OFFSET
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2,1
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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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}]
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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