The final 30 digits of 9999⁹, 359916012598740083996400089999, is prime in addition to having 9 copies of 9 and ending in 9999

Primes as sum of ascending powers in more than one way

Primes as a¹ + b² + c³ + d⁴ + e⁵  in more than one way:

139    

= 9¹ + 7² + 4³ + 2⁴ + 1⁵

= 14¹ + 9² + 3³ + 2⁴ + 1⁵ 

179      = 81 + 52 + 43 + 34 + 15

= 141 + 112 + 33 + 24 + 15 = 171 + 92 + 43 + 24 + 15

239    

= 12¹ + 9² + 4³ + 3⁴ + 1⁵  

= 13¹ + 7² + 4³ + 3⁴ + 2⁵

 = 14¹ + 12² + 4³ + 2⁴ + 1⁵

 = 16¹ + 9² + 5³ + 2⁴ + 1⁵ 

257

= 11¹ + 10² + 4³ + 3⁴ + 1⁵

= 14¹ + 6² + 5³ + 3⁴ + 1⁵ 

= 15¹ + 10² + 5³ + 2⁴ + 1⁵ 

= 16¹ + 8² + 4³ + 3⁴ + 2⁵ 

= 17¹ + 14² + 3³ + 2⁴ + 1⁵ 

571     = 141 + 122 + 53 + 44 + 25 = 151 + 102 + 73 + 34 + 25

= 151 + 142 + 73 + 24 + 15

= 171 + 92 + 63 + 44 + 15

= 171 + 152 + 63 + 34 + 25  Others primes as sum of powers (a1 + b2 + c3 + d4 +e5) in more than one way : 139, 157, 179, 181, 191, 193, 197, 199, 211, 223, 227, 239, 241, 251, 257, 269, 271, 281, 283, 293, 307, 311, 313, 331, 349, 359, 367, 373, 389, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 479, 487, 499, 521, 541, 547, 563, 569, 571, 593, 599, 617, 641, 673, 677, 719, 727, 739, 757, 761, 809, 937, 953, 971, 1097, 1103, 1129, 1201, 1297, 1327, 1423, 1777, 1979, 1997, 1999.

Products anagram

Since I was a kid, I have  always been amazed by the fact that when multiplying  four or seven by three, the two products obtained have the same digits but in a different position.
3 x 4 = 12
3 x 7 = 21

Now that I'm a little older, not much, it still surprises me that there are numbers that when they are multiplied by two different numbers, its products are a permutation of each other. Apparently you can find a number for each pair of distinct numbers provided that one of these numbers is not a multiple of ten of the other (ie for n and n * 10 ^ m, there is no number that when multiplied by a specific number, its products are not  anagrams).

Two years ago I published  8 sequences based on these facts in the OEIS. The title of each of these sequences is:  a(n) =smallest number such a(n)*n is an anagram of a(n)* X .

For example the sequence for X  equal to four is :

1782, 62937, 54, 1, 2826, 891, 3, 269, 631, 324, 2718, 4311, 3681, 37, 387, 25974, 4401, 477, 45, 48, 256437, 3393, 37, 26523, 3465, 3252, 3699, 34623, 2922, 27972, 27, 271, 284787, 27324, 25971, 263223, 26973, 25974, 2579247, 2514744     (OEIS A175693)

So:

1782   x 1 =      1782    and    1782 x 4 = 7218
62937 x 2 = 125874    and  62937 x 4 = 251748
54        x 3 =       162     and        54 x 4 = 216 
and so on.

Sometimes the same number meets the condition for example:
37 x 13 = 481, 37 x 22 = 814, 37 x 4 = 148

If we write these numbers in a table:

.		1	2	3	4	5	6	7	8	9
.	1	1	125874	1035	1782	142857	1386	1359	113967	1089
.	2	125874	1	1782	62937	5436	5175	774	891	9
.	3	1035	1782	1	54	36	41958	45	9	345
.	4	1782	62937	54	1	2826	891	3	269631	324
.	5	142857	5436	36	2826	1	9	279	252	2439
.	6	1386	5175	41958	891	9	1	693	27	594
.	7	1359	774	45	3	279	693	1	315	18
.	8	113967	891	9	269631	252	27	315	1	297
.	9	1089	9	345	324	2439	594	18	297	1
.
.		389350	202879	45265	338449	154135	50734	3487	385390	5116

we can see that even though the values ​​are different, oddly enough, the sum of the values ​​of number one is an anagram to the sum of the values ​​of the number eight: 389350 - 385390