Problems & Puzzles: Puzzles

Puzzle 988. Another puzzle about factorials Dmitry Kamenetsky sent the following nice puzzle What is the smallest factorial that can be represented as the sum of distinct primes raised to the power p? p=1: 4! = 24 = 11^1+13^1 p=2: 8! = 40320 = 2^2+3^2+5^2+7^2+11^2+13^2+17^2+19^2+23^2 +29^2+41^2+59^2+181^2 p=3: 10! = 3628800 = 5^3+19^3+29^3+37^3+47^3+151^3 p=4: 12! = 479001600 = 3^4+5^4+7^4+11^4+17^4+19^4+29^4+31^4+37^4+47^4+53^4+59^4 +73^4+79^4+97^4+131^4  Q. Send your best results for p>4.

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On the week from Feb 1 to Feb 7, contributions came from Michael Hürter and Dmitry Kamenetsky

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Michael  wrote:

I found the following solution for p = 5:

 

15! = 1307674368000 = 
2^5 +3^5 +7^5 +11^5 +17^5 +23^5 +29^5 +41^5 +43^5 +47^5 +53^5 +59^5 +61^5 +67^5 +97^5 +113^5 +131^5 +157^5 +167^5 +181^5 +191^5 +193^5 +197^5 

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Dmitry wrote:

I have a sequence for this puzzle: https://oeis.org/A308357

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From the sequence reported by Dmitry we can see two more solutions:

a(6) = 19, because 19! is the sum of the 6th powers of the primes in {3, 7, 17, 23, 37, 43, 47, 53, 61, 71, 73, 79, 89, 101, 103, 107, 113, 127, 137, 157, 167, 193, 211, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463}.

a(7) = 20, because 20! is the sum of the 7th powers of the primes in {5, 13, 31, 43, 59, 67, 71, 83, 97, 103, 109, 113, 137, 149, 167, 179, 181, 191, 193, 197, 227, 229, 233, 239, 241, 257, 263, 269, 277, 281, 283, 293, 311, 313, 317, 331}.

Note that these are the smallest k for which such a representation is possible.

We must note also that the a(5) solutions is a bit different than the obtained by Michael

a(5) = 15, because 15! = 13^5 + 17^5 + 19^5 + 31^5 + 37^5 + 41^5 + 53^5 + 61^5 + 89^5 + 97^5 + 139^5 + 163^5 + 199^5 + 241^5.

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