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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. 13
 3, 47, 151, 167, 199, 251, 257, 367, 557, 587, 601, 647, 727, 941, 971, 1097, 1117, 1181, 1217, 1361, 1499, 1741, 1747, 1901, 2281, 2411, 2671, 2897, 2957, 3301, 3307, 3631, 3727, 4007, 4397, 4451, 4591, 4651, 4679, 4987, 5101, 5107, 5297, 5381, 5387 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Subsets are A047948, A052188, A052189, A052190, A052195, A052197, A052198, etc. - R. J. Mathar, Apr 11 2008 Could be generated by searching for cases A001223(i) = A001223(i+1), writing down A000040(i). - R. J. Mathar, Dec 20 2008 a(n) = A006562(n) - A117217(n). - Zak Seidov, Feb 12 2013 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 LINKS Reinhard Zumkeller, Table of n, a(n) for n = 1..10000 FORMULA {A000040(i): A000040(i+1)= (A000040(i)+A000040(i+2))/2 }. - R. J. Mathar, Dec 20 2008 a(n) = A000040(A064113(n)). - Reinhard Zumkeller, Jan 20 2012 EXAMPLE 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. 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. MATHEMATICA 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 *) Transpose[Select[Partition[Prime[Range], 3, 1], #[] == (#[] + #[])/2 &]][]  (* Harvey P. Dale, Jan 09 2011 *) PROG (Haskell) a122535 = a000040 . a064113  -- Reinhard Zumkeller, Jan 20 2012 (PARI) A122535()={n=3; ctr=0; while(ctr<50, avgg=( prime(n-2)+prime(n) )/2; if( prime(n-1) ==avgg, ctr+=1; print( ctr, "  ", prime(n-2) )  ); n+=1); } \\ Bill McEachen, Jan 19 2015 CROSSREFS Cf. A006562, A062839, A102552, A117217, A181424. Sequence in context: A141850 A003551 A054643 * A058427 A142293 A052187 Adjacent sequences:  A122532 A122533 A122534 * A122536 A122537 A122538 KEYWORD nonn AUTHOR Miklos Kristof, Sep 18 2006 EXTENSIONS More terms from Roger L. Bagula, Nov 13 2008 Definition rephrased by R. J. Mathar, Dec 20 2008 STATUS approved

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Last modified November 20 14:54 EST 2019. Contains 329337 sequences. (Running on oeis4.)