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URL: https://oeis.org/A122535

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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.
24
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
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
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[750]], 3, 1], #[[2]] == (#[[1]] + #[[3]])/2 &]][[1]] (* 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
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