Problems & Puzzles: Puzzles

Puzzle 261.  Symmetric prime constellations

Someone named 'Dale' submitted the following curio to the well know Prime Curios! pages:

18731 is the smallest prime which is the average of both its immediate and second neighbors

That is to say, 18731, is at the 'center' of the following symmetrically spaced consecutive primes:

18713, 18719, 18731, 18743, 18749

such that 18731 = (18719+18743)/2=(18713+18749)/2

I have generalized this and ask for the earliest prime such that it is the average of its k-th immediate consecutive prime neighbors, for k=1 to m, for m=1, 2, 3, .... The Dale's example is the solution for m=2.

The sequence of these earlier primes goes like this: 5, 18731, 683783, 98303927, ... (?)

Question: Find more terms to this sequence.

Solution:

Faride Firoozbakht and J. K. Andersen found that the 5th term of this sequence was already discovered by Jud McCranie and reported in the EIS sequence A055380.

So the first real unknown term is the 6th one.

***

Giovanni Resta found it! (October, 2004):

...the next one (sixth of the sequence) is centered in 1169769749219 and the gaps are (single, cumulative): 6, 60, 12, 12, 6, 12, 12, 6, 12, 12, 60, 6 -108, -102, -42, -30, -18, -12, 0, 12, 18, 30, 42, 102, 108 so the constellation is:

1169769749327
1169769749321
1169769749261
1169769749249
1169769749237
1169769749231
1169769749219 <- center
1169769749207
1169769749201
1169769749189
1169769749177
1169769749117
1169769749111

***

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