Prime ensembles

LUC STEVENS , May 26 2006

• Introduction :

Take a prime number and make the sum of the digits. For some prime numbers that sum will be an other prime number. For example : s(43) = 4+3 = 7, which is also a prime number. ( s(n) stands for the sum of the digits of n. )

These two prime numbers are “related” to eachother and form a prime duet.

(numbers 43 and 2)

When we take the number 47 as an example, we will get s(47) = 4 + 7 = 11, but s(11) =

1 + 1= 2, another prime number. Now we have a prime trio (numbers 47,11 and 2)

• Definitions :

A prime duet is a pair of two different primenumbers such that the second number is a 1-digit number which is the sum of the digits of the first number.

A prime trio is a set of three different primenumbers such that the third number is a 1-digit number which is the sum of the digits of the second number, and the second number is the sum of the digits of the first number.

Likewise we can make prime quartets, prime quintets, etc…

Prime duets, prime trio’s, prime quartets, … are called prime ensembles.

The first number of each prime ensemble is called the “leader”. ( the largest number of the ensemble )

The last number of each ensemble must be 2,5 or 7.

• Prime ensemble leaders :

The first prime duetleaders are : (A119890)

11, 23, 41, 43, 61, 101, 113, 131, 151, 223, 241, 311, 313, 331, 401, 421, 601, 1013, 1031, 1033, 1051, 1103, 1123, 1213, 1231,1301, 1303, 1321, 2003, 2111, 2113, 2131, 2203, 2221, 2311, 3001, 3121, 3301, 4001, 4003, 4021, 4111, 4201, 5011, 5101, 10103, 10141, 10211, 10301, 10303, 10321, 10501, 11003, 11113, 11131, 11311, 12011, 12101, 12211, 12301, 13001, 13003, 14011

The first prime trio leaders are : (A119891)

29, 47, 83, 137, 173, 191, 227, 263, 281, 317, 353, 443, 461, 599, 641, 797, 821, 887, 911, 977, 1019, 1091, 1109, 1163, 1181, 1217, 1307, 1361, 1433, 1451, 1499, 1523, 1613, 1697, 1721, 1787, 1811, 1877, 1901, 1949, 2027, 2063, 2081, 2153, 2207, 2243, 2333, 2351, 2399, 2423, 2441, 2531, 2379, 2621, 2687, 2711, 2777, 2801, 2939, 2957, 3251, 3299, 3323, 3389, 3413, 3659, 3677, 3701, 3767, 3929, 3947, 4007

The first prime quartet leaders are : (A119892)

2999, 3989, 4799, 4889, 5879, 5897, 5987, 6599, 6689, 6779, 6869, 6959, 6977, 7499, 7589, 7877, 7949, 8597, 8669, 8849, 8867, 9479, 9497, 9587, 9677, 9749, 9767, 9839, 9857, 9929, 12899, 13799, 13997, 14699, 14879,14897,14969,15797,15887, 15959, 16787, 17489, 17579, 17597, 17669, 17939, 17957, 18587, 18749, 18839, 18947, 19289, 19379, 19469, 19559, 19577, 19739, 19793, 19919, 19937, 19973, 19991

• Each member, non-leader of a particular prime ensemble is the leader of one of the previous ensembles, with exception of the numbers 2,5 and 7.

Prime numbers which are no member of any prime ensemble are called a prime soloist.

Each primenumber must be soloist or a member of a prime ensemble.

The first prime soloists are : (A119889)

3, 13, 17, 19, 31, 37, 53, 59, 67, 71, 73, 79, 89, 97, 103, 107, 109, 127, 139, 149, 157, 163, 167, 179, 181, 193, 197, 199, 211, 229, 233, 239, 251, 257, 269, 271, 277, 283, 293, 307, 337, 347, 349, 359, 367, 373, 379, 383, 389, 397, 409, 419, 431, 433, 439, 449, 457, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569

• The smallest prime quintet leader is the smallest possible prime number for which the digital sum is 2999, if such number exists. If not, it can be a prime number with a digital sum of 3989 , etc…

The smallest possible number with a digital sum of 2999 is 299999…999 ( a 334-digit number beginning with 2 and followed by 333 9’s)

If this number is prime, it’s the smallest prime quintet leader.

Which number is the smallest prime quintet leader ?

• Thus, the sequence of the smallest prime ensemble leaders starts with 3 ( for the soloists), 11, 29, 2999, … ?

What is the next number in this sequence ?