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A120934 Least prime p such that the interval [p,p+log(p)] contains n primes. 3
2, 11, 457, 3251, 165701, 10526557, 495233351, 196039655873, 10687033762033, 79006533276941, 4313367040646743, 1740318019946551931 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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.
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.
LINKS
K. Soundararajan, The distribution of prime numbers, arXiv:math/0606408 [math.NT], 2006.
Eric Weisstein's World of Mathematics, k-Tuple Conjecture
FORMULA
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
A lower bound is a(n) > e^A008407(n). a(n) < b*e^A008407(n), for 2 <= n <= 12, b < 1.49. For 9 <= n <= 12, b < 1.0006. a(13) > 701673591209763173865. - Florian Baur, Jul 12 2023 [Corrected by Pontus von Brömssen, Nov 12 2023]
EXAMPLE
a(2)=11 because p=11 is the first prime with log(p) > 2 and 11+2 is prime.
MATHEMATICA
i=1; Table[While[p=Prime[i]; PrimePi[p+Log[p]]-PrimePi[p]+1< n, i++ ]; p, {n, 5}]
PROG
(PARI)
A120934(n) = {
my(v = vector(n, k, prime(k)), i = 1);
while(v[(i - 2) % n + 1] - v[i] > floor(log(v[i])),
v[i] = nextprime(v[(i - 2) % n + 1] + 1);
i = i % n + 1; );
return(vecsort(v)[1]) } \\ Florian Baur, Jul 12 2023
CROSSREFS
Cf. A120935, A120936 (number of primes in the interval [n, n+log(n)]), A020497.
Sequence in context: A012950 A012979 A013109 * A000886 A128855 A262547
KEYWORD
hard,more,nonn,nice
AUTHOR
T. D. Noe, Jul 21 2006
EXTENSIONS
a(12) from Donovan Johnson, Apr 18 2012
STATUS
approved

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Last modified April 23 19:56 EDT 2024. Contains 371916 sequences. (Running on oeis4.)