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A268593 The largest of three distinct primes, a > b > c > 0, such that each of the following four numbers are also prime: a+b+c, a+b-c, a-b+c, a-b-c, regardless of sign.

%I A268593 #30 Nov 18 2018 16:04:33
%S A268593 13,17,19,23,37,43,47,53,67,73,83,97,107,113,127,137,139,157,167,181,
%T A268593 193,197,199,223,227,229,233,251,263,269,277,283,317,337,347,349,373,
%U A268593 379,383,397,409,431,433,439,443,461
%N A268593 The largest of three distinct primes, a > b > c > 0, such that each of the following four numbers are also prime: a+b+c, a+b-c, a-b+c, a-b-c, regardless of sign.
%C A268593 A set of seven primes defined by the rules above may be called a "Prime Septet". For a given "a" value, from a(n) above, there may be multiple such septets.
%C A268593 These septets are found by searching all combinations of three distinct primes for a given candidate "a" value.
%C A268593 Once found, the following three properties of these septets are observed without exception for "a" values up to prime(800), resulting in these conjectures:
%C A268593   1.  a-b-c = +-3.
%C A268593   2.  b > a/2.
%C A268593   3.  5 <= c < a/2.
%C A268593   4.  If a prime p >= 3 belongs to one Prime Septet, then it belongs to an infinite number of Prime Septets.
%C A268593 From the first conjecture it follows that no values for "a", "b" or "c" belong to A172256.
%C A268593 The first three conjectures can be used to accelerate the search for prime septets. Further observations below are from accelerated searches.
%C A268593 There are a total of 43 "missing" values in a(n), compared to the full odd prime complement of A172256, starting with 3, 5, 7, 11, 29, 31, 41, 71, ...., 10677.
%C A268593 There are a total of 25 missing "b" values compared to the full odd prime complement of A172256. The last missing value is 3067.
%C A268593 It appears the values of "c" comprise the full odd prime complement of A172256.
%C A268593 The three sets of values for a+b+c, a+b-c and a-b+c, by contrast, split their membership between A172256 and its odd prime complement.
%C A268593 Positive vs. negative values for a-b-c occur with approximately equal frequency, with positive values at 48.05% of the total for all 1106 septets with a <= prime(1000); reaching 48.30% of the total for all 65821 septets with a <= prime(10000); and reaching the 48.97% of the total for all 195359 septets with a <= prime(20000). No other members of these septets are negative.
%C A268593 In the first 1000 primes, 502 primes play the role of an "a" value, with an average of 4 Prime Septets subordinate to that "a" value.
%C A268593 In the 1000 primes from prime(9001) to prime(10000), 376 primes play the role of an "a" value, with an average of 26 Prime Septets subordinate to each "a" value, with none having only 1,2,3,4 or 5 septets.
%C A268593 It appears the highest "a" value having only one subordinate septet is prime(2127)=18587, where {a,b,c} = {18587, 18427, 157}. This was checked for candidate "a" values up to prime(20000). By prime(10^6) it is typical for each "a" value to support 500 to 1500 Prime Septets.
%C A268593 It might seem that all odd primes should belong to at least one Prime Septet. This is not the case.
%C A268593 The primes not included in any Prime Septet may be called "Lonely Primes". The first few such odd primes are: 151, 179, 239, 293, 313. They are a subset of A172256.
%C A268593 Within the first 1000 odd primes there are 259 primes that are "Lonely". Within the first 10000 odd primes there are 3978 such primes. In the 1000 primes from prime(9001) to prime(10000), 455 of them are "Lonely".
%C A268593 It appears likely that "Lonely Primes" are a majority among all primes.
%e A268593 The first Prime Septet is based on {a,b,c} = {13,11,5}, with a+b+c =29,
%e A268593 a+b-c=19, a-b+c=7, a-b-c =-3. The first "a" value associated with more than one septet is 47, with {a,b,c} values of {47,37,13} and {47,43,7}.
%o A268593 (PARI) listp(nn) = {forprime(p=5, nn, ok = 0; forprime(q=3, p-1, forprime(r=2, q-1, if (isprime(abs(p+q+r)) && isprime(abs(p+q-r)) && isprime(abs(p-q+r)) && isprime(abs(p-q-r)), ok = 1; break);); if (ok, break);); if (ok, print1(p, ", ")););} \\ _Michel Marcus_, Mar 13 2016
%Y A268593 Cf. A000040, A172256.
%K A268593 nonn
%O A268593 1,1
%A A268593 _Richard R. Forberg_, Feb 07 2016