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A052187
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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.
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7
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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
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OFFSET
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1,1
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COMMENTS
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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
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LINKS
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FORMULA
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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.
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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