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A122535 Smallest prime of a triple of successive primes, where the middle one is the arithmetic mean of the other two.

%I A122535 #47 Mar 16 2024 18:18:13
%S A122535 3,47,151,167,199,251,257,367,557,587,601,647,727,941,971,1097,1117,
%T A122535 1181,1217,1361,1499,1741,1747,1901,2281,2411,2671,2897,2957,3301,
%U A122535 3307,3631,3727,4007,4397,4451,4591,4651,4679,4987,5101,5107,5297,5381,5387
%N A122535 Smallest prime of a triple of successive primes, where the middle one is the arithmetic mean of the other two.
%C A122535 Subsets are A047948, A052188, A052189, A052190, A052195, A052197, A052198, etc. - _R. J. Mathar_, Apr 11 2008
%C A122535 Could be generated by searching for cases A001223(i) = A001223(i+1), writing down A000040(i). - _R. J. Mathar_, Dec 20 2008
%C A122535 a(n) = A006562(n) - A117217(n). - _Zak Seidov_, Feb 12 2013
%C A122535 These are primes for which the subsequent prime gaps are equal, so (p(k+2)-p(k+1))/(p(k+1)-p(k)) = 1. It is conjectured that prime gaps ratios equal to one are less frequent than those equal to 1/2, 2, 3/2, 2/3, 1/3 and 3. - _Andres Cicuttin_, Nov 07 2016
%H A122535 Reinhard Zumkeller, <a href="/A122535/b122535.txt">Table of n, a(n) for n = 1..10000</a>
%H A122535 <a href="/index/Pri#primes_AP">Index entries for sequences related to primes in arithmetic progressions</a>
%F A122535 {A000040(i): A000040(i+1)= (A000040(i)+A000040(i+2))/2 }. - _R. J. Mathar_, Dec 20 2008
%F A122535 a(n) = A000040(A064113(n)). - _Reinhard Zumkeller_, Jan 20 2012
%e A122535 The prime 7 is not in the list, because in the triple (7, 11, 13) of successive primes, 11 is not equal (7 + 13)/2 = 10.
%e A122535 The second term, 47, is the first prime in the triple (47, 53, 59) of primes, where 53 is the mean of 47 and 59.
%t A122535 Clear[d2, d1, k]; d2[n_] = Prime[n + 2] - 2*Prime[n + 1] + Prime[n]; d1[n_] = Prime[n + 1] - Prime[n]; k[n_] = -d2[n]/(1 + d1[n])^(3/2); Flatten[Table[If[k[n] == 0, Prime[n], {}], {n, 1, 1000}]] (* _Roger L. Bagula_, Nov 13 2008 *)
%t A122535 Transpose[Select[Partition[Prime[Range[750]], 3, 1], #[[2]] == (#[[1]] + #[[3]])/2 &]][[1]]  (* _Harvey P. Dale_, Jan 09 2011 *)
%o A122535 (Haskell)
%o A122535 a122535 = a000040 . a064113  -- _Reinhard Zumkeller_, Jan 20 2012
%o A122535 (PARI) A122535()={n=3;ctr=0;while(ctr<50, avgg=( prime(n-2)+prime(n) )/2;
%o A122535 if( prime(n-1) ==avgg, ctr+=1;print( ctr,"  ",prime(n-2) )  );n+=1); } \\ _Bill McEachen_, Jan 19 2015
%Y A122535 Cf. A006562, A062839, A102552, A117217, A181424.
%K A122535 nonn
%O A122535 1,1
%A A122535 _Miklos Kristof_, Sep 18 2006
%E A122535 More terms from _Roger L. Bagula_, Nov 13 2008
%E A122535 Definition rephrased by _R. J. Mathar_, Dec 20 2008