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Primes pairs between ( (n/2) - ( log(n) )^2 ) and ( (n/2) + ( log(n) )^2 ) adding to even n

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Primes pairs between
((n / 2)  −  ( log n) 2 )
and
((n / 2) + ( log n) 2 )
adding to even
n

n
Prime less than
n / 2
are listed in decreasing order, primes from
n / 2
are listed in increasing order.
Distinct primes in pairs adding to
n
are surrounded by ■ (black squares); prime equal to
n / 2
surrounded by □ (white squares).
10 1
(4 primes)
(1.5 pairs)
Green tickY
2     ■3■

  □5□ ■7■
10 2
(11 primes)
(3 pairs)
Green tickY
■47■ 43 ■41■ 37 31 ■29■

■53■    ■59■ 61 67 ■71■
10 3
(14 primes)
(2 pairs)
Green tickY
499 ■491■ 487 ■479■ 467 463 461 457
 
503 ■509■     ■521■ 523 541     547
10 4
(23 primes)
(1 pair)
Green tickY
4999 4993 4987      4973 4969 4967 4957 4951 4943 4937 4933 4931      ■4919■

5003 5009 5011 5021 5023           5039 5051 5059                5077 ■5081■
10 5
(29 primes)
(2 pairs)
Green tickY
49999 49993 49991                   49957       49943 49939 49937 49927 49921 49919                   49891       ■49877■ ■49871■ 

                  50021 50023 50033 50047 50051 50053             50069 50077       50087 50093 50101 50111 50119 ■50123■ ■50129■ 50131 
10 6
(27 primes)
(1 pair)
Green tickY
       499979 499973 499969 499957 ■499943■ 499927 499903 499897               499883 499879 499853                      499819

500009        500029        500041 ■500057■ 500069 500083 500107 500111 500113 500119        500153 500167 500173 500177 500179
10 9
(41 primes)
(2 pairs)
Green tickY
  
          499999993                     ■499999931■                     499999909                     499999897 499999873 499999853  499999847  499999831 
                                         499999751  499999723 499999697 499999693                                                   ■499999613■
 

500000003 500000009 500000041 500000057 ■500000069■ 500000071 500000077 500000089 500000093 500000099 500000101 500000117                       500000183 
500000201 500000227 500000231 500000233  500000261  500000273 500000299 500000317 500000321 500000323 500000353 500000359 500000377 ■500000387■ 500000393
10 12
(52 primes)
(2 pairs)
Green tickY
499999999979 499999999943 499999999901 499999999897 499999999847 499999999819              499999999799 ■499999999769■              499999999739 499999999699 499999999661  499999999643  499999999571 
499999999559 499999999511 499999999507 499999999501 499999999487 499999999451 499999999427 499999999403  499999999391                            499999999357              ■499999999277■

500000000023 500000000033 500000000089 500000000131 500000000147 500000000173 500000000191 500000000209 ■500000000231■ 500000000243 500000000263 500000000273 500000000333  500000000387  500000000413 
500000000471                           500000000509 500000000537 500000000551              500000000609  500000000611  500000000623 500000000627 500000000651 500000000677 ■500000000723■ 500000000737 500000000761

1477665855854
(2⋅738832927927)
(48 primes)
(1.5 pair)
Green tickY
 738832927913 738832927909 738832927883 738832927879 738832927873 738832927867 738832927829 738832927799 738832927753 738832927711 738832927691 738832927681 
 738832927673 738832927663 738832927631 738832927607 738832927597 738832927579 738832927559 738832927543 738832927523 738832927499 738832927477 
■738832927387■ 
 738832927373 738832927367 738832927363 738832927331 738832927277 738832927271 738832927267 738832927253 738832927247 738832927201 738832927193 738832927187 738832927169 738832927153 

□738832927927□<!-- gap of 540 (see A053303) from 738832927927 (see A053302) to 738832928467 -->
   
■738832928467■ 
 738832928473 738832928507 738832928569 738832928593 738832928621 738832928623 738832928629 738832928683

Cramér’s conjecture

Consider Cramér’s conjecture (by the Swedish mathematician Harald Cramér in 1936, yet unproved)

pn  +1pn
(log pn ) 2
= 1,
where
pn
is the
n
-th prime.
In the [rare] cases where a primefree interval of width near
( log pn ) 2
falls somewhere in the middle of
  • ((n / 2)  −  ( log n) 2 )
    to
    ((n / 2) + ( log n) 2 )
there will be a low probability of having prime pairs adding to even
n
.
In the [much rarer] cases where a primefree interval of width near
( log pn ) 2
almost coincides with either
  • ((n / 2)  −  ( log n) 2 )
    to
    n / 2
    , or
  • n / 2
    to
    ((n / 2) + ( log n) 2 )
    ,
there will be a near zero probability of having prime pairs adding to even
n
.
It thus appears that, asymptotically, it will happen infinitely often that there will be no pair of distinct primes within
((n / 2)  −  ( log n) 2 )
and
((n / 2) + ( log n) 2 )
adding to even
n
.

Question:

  • Is the asymptotic density of cases where no pair of distinct primes within
    ((n / 2)  −  ( log n) 2 )
    and
    ((n / 2) + ( log n) 2 )
    adding to even
    n
    equal to 0?

Sequences

A053303 Length of maximal prime gap
pk  +1  −  pk
with starting prime
pk < 10n
.
{4, 8, 20, 36, 72, 114, 154, 220, 282, 354, 464, 540, 674, 804, 906, 1132, ...}
A053302 Largest
n
-digit prime at the start of a record in the RECORDS transform of the prime gaps.
{7, 89, 887, 9551, 31397, 492113, 4652353, 47326693, 436273009, 4302407359, 42652618343, 738832927927, 7177162611713, 90874329411493, 218209405436543, 1693182318746371, ...}

External links