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A120934 Least prime p such that the interval [p,p+log(p)] contains n primes. 2


%S 2,11,457,3251,165701,10526557,495233351,196039655873,10687033762033,

%T 79006533276941,4313367040646743,1740318019946551931

%N Least prime p such that the interval [p,p+log(p)] contains n primes.

%C Soundararajan states that, on average, there is one prime in the interval [k,k+log(k)] for any number k. Is there an upper limit to the number of primes in such an interval? Not if the prime k-tuple conjecture is true, in which case a(n) exists for all n. Note that a(n) > e^A008407(n). See A120935 for the largest prime in the interval.

%C a(n) begins a sequence of n primes whose prime pattern is one of the patterns in the n-th row of A186634. For example, the sequence of four consecutive primes beginning with 3251 is (3251, 3253, 3257, 3259), which has pattern (0, 2, 6, 8), which is in the 4th row of A186634.

%H K. Soundararajan, <a href="http://www.arXiv.org/abs/math.NT/0606408">The distribution of prime numbers</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/k-tupleCojecture.html">Prime k-Tuple Conjecture</a>

%F This sequence grows superexponentially; a weak lower bound is a(n) >> (log n)^n. It seems that a(n) > n^n. - _Charles R Greathouse IV_, Apr 18 2012

%e a(2)=11 because p=11 is the first prime with log(p)>2 and 11+2 is prime.

%t i=1; Table[While[p=Prime[i]; PrimePi[p+Log[p]]-PrimePi[p]+1< n, i++ ]; p, {n,5}]

%Y Cf. A120936 (number of primes in the interval [n, n+log(n)]), A020497.

%K hard,nonn,nice

%O 1,1

%A _T. D. Noe_, Jul 21 2006

%E a(12) from _Donovan Johnson_, Apr 18 2012

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Last modified August 25 13:50 EDT 2019. Contains 326324 sequences. (Running on oeis4.)