2019

Sure, the number 2019 has a lot of great properties. For example, 2019 is the least number having six different representations as a sum of three squares of prime numbers.

2019 = 23²+23²+31²

2019 = 17²+19²+37²

2019 = 11²+23²+37²

2019 = 13²+13²+41²

2019 = 7²+17²+41²

2019 = 7²+11²+43²

There a lot of numbers having six different representations as above, not seven or more. They form a sequence 2019, 2091, 2499, 3099, 3219, 3339, 3579, 3939, 4011, ... E.g., 4011 = 31² + 37² + 41² = 29² + 31² + 47² = 11² + 41² + 47² = 19² + 29² + 53² = 13² + 19² + 59² = 11² + 13² + 61².

Puzzle 1: Sum of Powers of Primes / Results - 2

Dana Jacobsen has published a short code for (P2) in the Perl language on mersenneforum. He notes that testing up to 10⁹ takes 90 minutes.

Puzzle 1: Sum of Powers of Primes / Results - 1

The mersenneforum user, nicknamed a1call, has announced the finding of a new number with the (P2) property. It is the number 9634877 (Sep 3rd, 2018). Excellent work!

Puzzle 1: Sum of Powers of Primes

The whole story began around the turn of the millennium when Carlos Rivera  observed that 39 is a number with the following property:

(P1)     A non-prime number equal to the sum of consecutive primes from the least prime factor to the largest prime factor.

Of course, 39 = 3 * 13 = 3 + 5 + 7 + 11 + 13.

It is not difficult to verify that 10, 155 and 371 are other numbers with this property. Later (Jul. 2000), Jud McCranie found the 5th number:

2935561623745 = 5 * 19 * 53 * 61 * 9557877 = 5 + 7 + 11 + ... + 9557877.

On Nov. 2002, Robert Munafo discovered that

454539357304421 = 3536123 * 128541727 = 3536123 + 3536129 + 3536131 + ... + 128541727.

So, six numbers with the property (P1) are known up to now. One can find them in the Online Encyclopedia of Integer Sequences under the code A055233.

We change the property somewhat now:

(P2)     A non-prime-square number equal to the sum of squares of consecutive primes from the least prime factor to the largest prime factor.

The smallest number with this property is 315797.

Indeed,

315797 = 31 * 61 * 167 = 31² + 37² + 41² + ... + 167².

Question A: Find other numbers with the property (P2).

We can continue.

(P3)     A non-prime-cube number equal to the sum of cubes of consecutive primes from the least prime factor to the largest prime factor.

In this case, there is a nice solution: 160.

160 = 2⁵ * 5 = 2³ + 3³ + 5³.

Question B: Find other numbers with the property (P3).

And finally:

Question C: The first, second and third powers of primes can be generalized to any n-th power. Find also any numbers for n>3.