A089637 Smallest member of a pair of consecutive twin prime pairs that have exactly n primes between them.

3, 17, 41, 107, 71, 2267, 1091, 461, 1319, 1151, 347, 5741, 2999, 5279, 10139, 1487, 9461, 881, 659, 13007, 9041, 15359, 8627, 28751, 83717, 13397, 18539, 14627, 44771, 54011, 60257, 59669, 142157, 77711, 61559, 178931, 26261, 122867, 293261, 89069, 24419, 167861

Offset: 0

Author

Cino Hilliard, Jan 01 2004

Keywords

• nonn

Comments

Smallest prime p such that n primes exist between the twin prime pair (p, p+2) and the next twin prime pair.

If this sequence is well defined then the Twin Prime Conjecture is true. - David A. Corneth, Dec 27 2019

Examples

			a(0) = 3 since there is no prime between the twin primes (3, 5) and (5, 7). - _David A. Corneth_, Dec 27 2019
a(1) = 17 since there is one prime, 23, between the twin primes (17, 19) and (29, 31).
a(2) = 41 since there are 2 primes, 47 and 53, between the twin primes (41, 43) and (59, 61).
		

Links

• David A. Corneth, Table of n, a(n) for n = 0..342 (terms n = 1..226 from and terms > 10^12 from Amiram Eldar)
• David A. Corneth and Amiram Eldar, Terms <= 1.5*10^12 (0 indicates the term is > 1.5*10^12) (terms > 10^12 from Amiram Eldar)

Crossrefs

Cf. A001097, A001359, A006512, A027833.

Programs

Maple

A181981 := proc(n)
    local j,hi,lo ;
    if n = 0 then
        3;
    else
        for j from 1 do
                hi := numtheory[pi](A001359(j+1)) ;
                lo := numtheory[pi](A006512(j)) ;
                if hi-lo = n+1 then
                    return A001359(j);
                end if;
        end do:
    end if;
end proc: # R. J. Mathar, Jul 03 2012

Mathematica

countPrimes[pin_] := Module[{prv = pin, c = 0, p}, p = NextPrime[prv]; While[p != prv + 2, c++; prv = p; p = NextPrime[p]]; {c-1, p}]; p = 13; mx = 20; c = 0; seq = Table[0, {mx}]; While[c < mx, cp = countPrimes[p]; i = cp[[1]]; If[i > 0 && i <= mx && seq[[i]] == 0, c++; seq[[i]] = p - 2]; p = cp[[2]]]; seq (* Amiram Eldar, Dec 26 2019 *)

PARI

pbetweentw(n) = /* p is the number of primes between */ { for(p=0, 100, forstep(x1=1, n, 1, my(c=0, t1 = twin[x1], t2 = twin[x1+1]); for(y=t1+4, t2-1, if(isprime(y), c++) ); if(c==p, print1(t1", "); break) ) ) }
savetwins(n) = /* Build a twin prime table of lower bounds */ { twin = vector(n); my(c=1); forprime(x=3, n*10, if(isprime(x+2), twin[c]=x; c++; ) ) }

Extensions

Offset corrected and data corrected and expanded by Amiram Eldar, Dec 26 2019

a(0) = 3 prepended by David A. Corneth, Dec 27 2019