A186634 -id:A186634 - cp's OEIS frontend

A186702 Irregular triangle, read by rows, of the smallest prime starting a prime pattern given in row n of A186634.

3, 5, 7, 5, 5, 7, 7, 11, 5639, 11, 17, 88793, 11, 17, 13, 88789, 11, 9853497737, 11, 1418575498573, 11, 1418575498567, 11, 7697168877290909, 10527733922579, 1707898733581273, 13, 186460616596321, 11, 79287805466244209, 11, 17, 1158722981124148367, 14094050870111867483, 47710850533373130107, 13, 17, 13, 734975534793324512717947, 1620784518619319025971, 13, 2845372542509911868266807

Offset: 2

T. D. Noe, Feb 25 2011

The larger terms are from the Forbes file.

			5639 is the smallest prime that begins a sequence of 7 primes having pattern (0, 2, 8, 12, 14, 18, 20).
The irregular triangle begins:
   3
   5           7
   5
   5           7
   7
  11        5639
  11          17  88793
  11          17     13  88789
  11  9853497737

Tony D. Forbes, Smallest Prime k-tuples

Cf. A186634.

A374000 a(n) = Product_{i=1..m} prime(k + T(n,i)) where k = pi(A186702(n)), T(n,i) is the i-th term in row n of A186634, and m = length of row n of A186634.

Original entry on oeis.org

15, 385, 1001, 5005, 85085, 323323, 7436429, 955049953, 183698727318433150098859517, 35336848261, 435656388001, 3868985835982814590518552822749329543261, 1448810778701, 20475850236047, 5663533044013, 343523383391078124677551786579090220816600929, 62298863484143

Offset: 1

Michael De Vlieger, Jul 04 2024

nonn

			Let p = A186702 and let T(n,i) be the i-th term in row n of A186634.
a(1) = 15 since p(1) = 3 and row 1 of T is {0, 2}, hence 3 * (3+2) = 3 * 5 = 15.
a(2) = 385 since p(2) = 5 and row 2 of T is {0, 2, 4}, hence 5 * (5+2) * (5+2+4) = 5*7*11 = 385.
Prime decomposition of the first 8 terms.
        a(n)    k  k+m-1  prime decomposition.
----------------------------------------------
         15     2     3    3 *  5
        385     3     5    5 *  7 * 11
       1001     4     6    7 * 11 * 13
       5005     3     6    5 *  7 * 11 * 13
      85085     3     7    5 *  7 * 11 * 13 * 17
     323323     4     8    7 * 11 * 13 * 17 * 19
    7436429     4     9    7 * 11 * 13 * 17 * 19 * 23
  955049953     5    11   11 * 13 * 17 * 19 * 23 * 29 * 31

Cf. A005117, A120944, A186634, A186702.

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

Original entry on oeis.org

2, 11, 457, 3251, 165701, 10526557, 495233351, 196039655873, 10687033762033, 79006533276941, 4313367040646743, 1740318019946551931

Offset: 1

T. D. Noe, Jul 21 2006

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.

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

K. Soundararajan, The distribution of prime numbers, arXiv:math/0606408 [math.NT], 2006.
Eric Weisstein's World of Mathematics, k-Tuple Conjecture

Cf. A120935, A120936 (number of primes in the interval [n, n+log(n)]), A020497.

Cf. A008407, A186634.

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

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

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]

a(12) from Donovan Johnson, Apr 18 2012

cp's OEIS Frontend

A186702 Irregular triangle, read by rows, of the smallest prime starting a prime pattern given in row n of A186634.

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A374000 a(n) = Product_{i=1..m} prime(k + T(n,i)) where k = pi(A186702(n)), T(n,i) is the i-th term in row n of A186634, and m = length of row n of A186634.

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

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