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.

13, 17, 19, 23, 37, 43, 47, 53, 67, 73, 83, 97, 107, 113, 127, 137, 139, 157, 167, 181, 193, 197, 199, 223, 227, 229, 233, 251, 263, 269, 277, 283, 317, 337, 347, 349, 373, 379, 383, 397, 409, 431, 433, 439, 443, 461

Offset: 1

Author

Richard R. Forberg, Feb 07 2016

Keywords

• nonn

Comments

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.

These septets are found by searching all combinations of three distinct primes for a given candidate "a" value.

Once found, the following three properties of these septets are observed without exception for "a" values up to prime(800), resulting in these conjectures:

1. a-b-c = +-3.

2. b > a/2.

3. 5 <= c < a/2.

4. If a prime p >= 3 belongs to one Prime Septet, then it belongs to an infinite number of Prime Septets.

From the first conjecture it follows that no values for "a", "b" or "c" belong to A172256.

The first three conjectures can be used to accelerate the search for prime septets. Further observations below are from accelerated searches.

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.

There are a total of 25 missing "b" values compared to the full odd prime complement of A172256. The last missing value is 3067.

It appears the values of "c" comprise the full odd prime complement of A172256.

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.

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.

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.

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.

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.

It might seem that all odd primes should belong to at least one Prime Septet. This is not the case.

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.

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".

It appears likely that "Lonely Primes" are a majority among all primes.

Examples

			The first Prime Septet is based on {a,b,c} = {13,11,5}, with a+b+c =29,
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}.
		

Crossrefs

Cf. A000040, A172256.

Programs

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