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A089637 Smallest member of a pair of consecutive twin prime pairs that have exactly n primes between them. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

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

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)

EXAMPLE

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).

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 *)

PROG

(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++; ) ) }

CROSSREFS

Cf. A001097, A001359, A006512.

Sequence in context: A209544 A146821 A181981 * A135471 A322755 A226492

Adjacent sequences:  A089634 A089635 A089636 * A089638 A089639 A089640

KEYWORD

nonn

AUTHOR

Cino Hilliard, Jan 01 2004

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

STATUS

approved

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Last modified June 30 19:43 EDT 2022. Contains 354945 sequences. (Running on oeis4.)