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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.
7
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
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