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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.

%I A052187 #60 Jun 22 2025 03:55:10
%S A052187 3,47,199,20183,16763,69593,255767,247099,3565931,6314393,4911251,
%T A052187 12012677,23346737,43607351,34346203,36598517,51041957,460475467,
%U A052187 652576321,742585183,530324329,807620651,2988119207,12447231761,383204539,4470607951,5007182707
%N 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.
%C A052187 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.
%C A052187 a(54) > 5*10^13, while a(55) = 46186474937633. - _Giovanni Resta_, Apr 08 2013
%H A052187 Jerry M Lagrou, <a href="/A052187/b052187.txt">Table of n, a(n) for n = 1..72</a> (terms 1..39 from Donovan Johnson, terms 40..53 from Giovanni Resta)
%F A052187 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.
%F A052187 a(1) = A054342(1) - 2. For n>1, a(n) = A054342(n) - 6*(n-1). - _Jeppe Stig Nielsen_, Apr 16 2022
%e A052187 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.
%t A052187 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, 10^6}]; a (* Typos fixed by _Zak Seidov_, May 01 2020 *)
%o A052187 (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
%Y A052187 Cf. A001223, A047948, A052160, A054342.
%Y A052187 Cf. A052188, A052189, A052195, A052196, A052197, A052198.
%K A052187 nonn
%O A052187 1,1
%A A052187 _Labos Elemer_, Jan 28 2000
%E A052187 More terms from _Labos Elemer_, Jan 04 2002
%E A052187 More terms from _Robert G. Wilson v_, Jan 06 2002
%E A052187 Definition clarified by _Harvey P. Dale_, Aug 29 2012
%E A052187 a(23)-a(27) from _Donovan Johnson_, Aug 30 2012
%E A052187 Name edited by _Jon E. Schoenfield_, Nov 30 2023