A030173 Differences p(i)-p(j) between primes, sorted in numerical order.

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 20, 21, 22, 24, 26, 27, 28, 29, 30, 32, 34, 35, 36, 38, 39, 40, 41, 42, 44, 45, 46, 48, 50, 51, 52, 54, 56, 57, 58, 59, 60, 62, 64, 65, 66, 68, 69, 70, 71, 72, 74, 76, 77, 78, 80, 81, 82, 84, 86, 87, 88, 90

Offset: 1

Author

Alexander Grasser [Graesser] (alex(AT)computicket.com)

Keywords

• nonn
• easy
• nice

Comments

Conjectured (Polignac 1849) to be union of even numbers and the odd primes minus 2.

For n > 2: A092953(a(n)) > 0. - Reinhard Zumkeller, Nov 10 2012

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• K. Soundararajan, Small gaps between prime numbers: the work of Goldston-Pintz-Yildirim, Bull. Amer. Math. Soc., 44 (2007), 1-18.
• Index entries for primes, gaps between

Crossrefs

Complement of A007921. Cf. A001223, A005843, A040976.

Programs

Haskell

import Data.List.Ordered (union)
a030173 n = a030173_list !! (n-1)
a030173_list = union [2, 4 ..] $ tail a040976_list
-- Reinhard Zumkeller, Jul 03 2015

Mathematica

nn = 90; Union[Range[2, nn, 2], Prime[Range[2, PrimePi[nn+2]]] - 2]

PARI

print1(1);p=3;forprime(q=5,1e3,forstep(n=p-1,q-3,2,print1(", "n));print1(", ",q-2);p=q) \\ conjectural; Charles R Greathouse IV, Jul 02 2011

PARI

isOK(n)=if(n%2,isprime(n+2),forprime(p=3,,isprime(n+p)&&return(1)));
for(n=1,10^100,isOK(n)&print1(n,", ")) \\ unconditionally outputs correct values only, will "hang" forever if conjecture is false once that exceptional even number is reached; Jeppe Stig Nielsen, Sep 23 2015