A394299 a(n) is the numerator of the expected absolute difference of two primes p1, p2 drawn randomly from the interval 2^n <= p < 2^(n+1) without replacement.

1, 2, 2, 38, 236, 850, 1000, 26840, 477368, 47036, 22339372, 73096731, 260586441, 1180973647, 3853501438, 178350939014, 1261487067476, 3639250753600, 26083459413092, 946868742489918, 264892551871161, 5030764357055773

Offset

1,2

Links

Table of n, a(n) for n=1..22.

Example

1/1, 2/1, 2/1, 38/5, 236/21, 850/39, 1000/23, 26840/301, 477368/2775, 47036/137, ...
a(4)/A394300(4) = 38/5: The 5 primes in the interval [16,31] are 17, 19, 23, 29, 31. Their A036378(4)*(A036378(4)-1)/2 = (5*4)/2 = 10 mutual differences are 2, 6, 12, 14, 4, 10, 12, 6, 8, 2, with sum 76.
If all numbers from the interval were in the urn, there would be A006516(4) = 120 pairs, with a mean of A000051(4)/3 = (2^4 + 1)/3 = 17/3. The following table shows that the naive assumption that the expected value of the differences of the drawn prime number pairs is always larger than that obtained when all numbers are in the urn is incorrect.
.
   n      A394299(n)/A394300(n)    rounded     (2^n+1)/3
   2               2/1               2.0          1.667
   3               2/1               2.0    <     3.0
   4              38/5               7.6          5.667
   5             236/21             11.238       11.0
   6             850/39             21.795       21.667
   7            1000/23             43.478       43.0
   8           26840/301            89.169       85.667
   9          477368/2775          172.025      171.0
  10           47036/137           343.328      341.667
  11        22339372/32385         689.806      683.0
  12        73096731/53708        1361.002 <   1365.667
  14      1180973647/216411       5457.087 <   5461.667
  15      3853501438/352995      10916.589 <  10923.0
  17   1261487067476/28882563    43676.424 <  43691.0
  19  26083459413092/149262459  174748.960 < 174763.0

Prog

(PARI) a394299_300(n) = my(n2=2^n, n22=n2+n2, s=0, k=0); forprime(p1=n2, precprime(precprime(n22)-1), forprime(p2=nextprime(p1+1), n22, s+=(p2-p1); k++)); s/k

Crossrefs

A394300 are the corresponding denominators.

Cf. A000051, A006516, A036378.

Sequence in context: A059523 A038623 A001121 * A202711 A222668 A210246

Adjacent sequences: A394296 A394297 A394298 * A394300 A394301 A394302

Keyword

nonn,frac,more

Author

Hugo Pfoertner, Mar 23 2026

Status

approved