A052187 a(n) is the smallest prime p such that p, p+d, and p+2d are consecutive primes where d = 2 for n = 1 and d = 6*(n-1) for n > 1.

3, 47, 199, 20183, 16763, 69593, 255767, 247099, 3565931, 6314393, 4911251, 12012677, 23346737, 43607351, 34346203, 36598517, 51041957, 460475467, 652576321, 742585183, 530324329, 807620651, 2988119207, 12447231761, 383204539, 4470607951, 5007182707

Offset

1,1

Comments

The first term 3 is anomalous since for all others d is divisible by 6. These are minimal terms if in A047948 d=6 is replaced by possible differences: (2), 6, 12, 18, ..., 54, 60.

a(54) > 5*10^13, while a(55) = 46186474937633. - Giovanni Resta, Apr 08 2013

Links

Jerry M Lagrou, Table of n, a(n) for n = 1..72 (terms 1..39 from Donovan Johnson, terms 40..53 from Giovanni Resta)

Formula

The least prime(k) such that prime(k+1) = (prime(k) + prime(k+2))/2 and prime(k+1) - prime(k) = d is either 2 or divisible by 6.

a(1) = A054342(1) - 2. For n>1, a(n) = A054342(n) - 6*(n-1). - Jeppe Stig Nielsen, Apr 16 2022

Example

a(2)=47 and it is the lower border of a dd pattern: 47[6 ]53[6 ]59. a(10)=6314393 and a(10)+54=6314447, a(10)+108=6314501 are consecutive primes and 6314393 is the smallest prime prior to a (54,54) difference pattern of A001223.

Mathematica

a = Table[0, {100}]; NextPrime[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; p = q = r = 0; Do[r = NextPrime[r]; If[r + p == 2q && r - q < 201 && a[[(r - q)/2]] == 0, a[[(r - q)/2]] = p; p = q; q = r, {n, 1, 10^8}]; a (* Typos fixed by Zak Seidov, May 01 2020 *)

Prog

(PARI) list(n)=ve=vector(n); ppp=2; pp=3; forprime(p=5, , d=p-pp; if(pp-ppp==d, i=d\6+1; if(i<=n&&ve[i]==0, ve[i]=ppp; print1("."); vecprod(ve)>0&&return(ve))); ppp=pp; pp=p) \\ Jeppe Stig Nielsen, Apr 17 2022

Crossrefs

Cf. A001223, A047948, A052160, A054342.

Cf. A052188, A052189, A052195, A052196, A052197, A052198.

Sequence in context: A122535 A058427 A142293 * A260219 A131465 A277388

Adjacent sequences: A052184 A052185 A052186 * A052188 A052189 A052190

Keyword

nonn

Author

Labos Elemer, Jan 28 2000

Extensions

More terms from Labos Elemer, Jan 04 2002

More terms from Robert G. Wilson v, Jan 06 2002

Definition clarified by Harvey P. Dale, Aug 29 2012

a(23)-a(27) from Donovan Johnson, Aug 30 2012

Name edited by Jon E. Schoenfield, Nov 30 2023

Status

approved