23 It is the smallest number that is the concatenation of two primes 2 and 3.
237 is the smallest number that can be separated into two primes in two different ways: 2-37, 23-7
2337 is the smallest number that can be separated into two primes in three different ways: 2-337, 23-37, 233-7
29397 is the smallest number that can be separated into two primes in four different ways: 2-9397, 29-397, 293-97, 2939-7
Hence we have the sequence 23, 237, 2337, 29397
What is the next term in this sequence?
The smallest square that can be divided into two primes is 25
52 = 25 and 25 can be divided into two primes 2-5
772 = 5929
and this is the smallest square that can be divided into two primes in two different ways: 59-29 and 5-929
15492 = 2399401 and this is the smallest square that can be divided into two primes in three different ways: 2399-401, 23-99401 and 2-399401.
230772 = 532547929 and this is the smallest square that can be divided into two primes in four different ways: 5-32547929 ,
53-2547929 ,
532547-929 and
5325479-29
Hence we have the sequence 5, 77, 1549, 23077
What is the next term in this sequence?
domingo, 3 de junio de 2012
viernes, 18 de mayo de 2012
lunes, 27 de febrero de 2012
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. |
jueves, 26 de enero de 2012
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:
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