Building Sequences from a Seed Pair

 FIRST EXAMPLE

The numbers 1 and 4 have the interesting properties that:

• their sum is prime: 1 + 4 = 5
• their difference is prime: 4 - 1 = 3
• their product (4) is the average of a pair of twin primes (3 and 5)

Let's make this the starting point of a sequence and let the third member of the sequence be \(x\). This gives us: \(1, 4, x\). We want 4 and \(x\) to share the properties that 1 and 4 enjoyed. Namely:

• 4 + \(x\) is prime
• \(x\) - 4 is prime
• 4 \(\times x\) is the average of a pair of twin primes

A little trial and error shows that the smallest value of \(x\) we are looking for is 15 because:

• 4 + 15 = 19 is prime
• 15 - 4 = 11 is prime
• 4 \( \times \) 15 = 60 is average of a pair of twin pairs (59 and 61)

By using not trial and error but a simple algorithm we can find further terms. The result is OEIS A154493 and the initial terms are:

1, 4, 15, 28, 39, 50, 81, 350, 459, 512, 675, 944, 987, 1040, 1917, 1936, 2325, 2378, 2421, 2588, 2745, 2812, 3459, 3488, 3495, 3506, 5667, 5804, 6027, 6074, 24765, 24832, 25479, 25552, 27621, 27848, 27951, 27980, 34101, 34720, 34773, 35344

SECOND EXAMPLE

Let's take another seed pair with the simple property that the two numbers must add to a cubic number. We'll use 1 and 7 as our seed pair because: \(1 + 7 = 8 = 2^3\). Here we see that the next number must be 20 because \(7+20=27=3^3\). Using an algorithm, the sequence generated begins thus (permalink):

1, 7, 20, 44, 81, 135, 208, 304, 425, 575, 756, 972, 1225, 1519, 1856, 2240, 2673, 3159, 3700, 4300, 4961, 5687, 6480, 7344, 8281, 9295, 10388, 11564, 12825, 14175, 15616, 17152, 18785, 20519, 22356, 24300, 26353, 28519, 30800, 33200, 35721, 38367

THIRD EXAMPLE

Let's start with seed numbers 1 and 2 this time with the property that:

• the sum of the two numbers has a digit sum that is prime
• the product of the two numbers plus 1 has a digit product that is prime

The seed pair 1 and 2 satisfy since:

• the sum of 1 and 2 is 3 and 3 is prime
• the product of 1 and 2 plus 1 is 3 and 3 is prime

This leads to the following sequence: 1, 2, 9, 12, 13, 16, 18, 23, 24, 25, 27, 29, 32, 33, 34, 40, 45, 47, 51, 60, 62, 66, 100, ... (permalink). There are 2211 terms in the range up to 40000.