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

The name of the numbers in English begins with one of these 13 letters : O,T,F,S,E,N,H,M,B,Q,U,D or G
So, the smallest numbers that start with different N letters in English are :

1                         1 One
2                       21 Twenty One
3                     102 One Hundred Two
4                     124 One Hundred Twenty Four
5                  1.146 One Thousand one Hundred Forty Six
6                  1.468 One Thousand Four Hundred Sixty Eight
7                41.689 Forty One Thousands Six Hundred Eighty Nine
8           1.004.689 One Million Four Thousands Six Hundred Eighty Nine
9    1.002.040.689 One Billion Two Millions Forty thousands Six Hundred Eighty Nine
10                          One Quadrillion one Billion Two Millions Forty thousands Six Hundred Eighty Nine
11                          One Undecillion one Quadrillion one Billion Two Millions Forty thousands Six Hundred Eighty Nine
12                          One Duodecillion one Undecillion one Quadrillion one Billion Two Millions Forty Thousands Six Hundred Eighty Nine
13                          One Googol one Duodecillion one Undecillion one Quadrillion one Billion Two Millions Forty Thousands Six Hundred Eighty Nine

Numbers and letters

Write the natural numbers in English : one, two, three, four, etc


One has the first o, the first n, and the first e
Two has the second o
Three has the third e


And :


11 has the 11 e 
(onE two thrEE four fivE six sEvEn Eight NinE tEn ElEvEn)
23 has the 23 t
24 has the 24 t
29 has the 29 n
31 has the 31 n
108 has the 108 n
109 has the 109 n
198 has the 198 d
199 has the 199 d
240 has the 240 r
241 has the 241 r
243 has the 243 h
244 has the 244 h
245 has the 245 h
246 has the 246 h
247 has the 247 h
248 has the 248 h
249 has the 249 o
250 has the 250 o
251 has the 251 o
453 has the 453 u
454 has the 454 u
559 has the 559 i
1174 has the 1174 o
1716 has the 1716 s
5556 has the 5556 f
5557 has the 5557 f
6956 has the 6956 f
6957 has the 6957 f
15756 has the 15756 f
17155 has the 17155 f
24998 has the 24998 y
24999 has the 24999 y
43568 has the 43568 f
43569 has the 43569 f
735759 has the 735759 v
1105805 has the 1105805 v
1105806 has the 1105806 v
1105807 has the 1105807 v
1107784 has the 1107784 v
1107785 has the 1107785 v
1584503 has the 1584503 v
1584504 has the 1584504 v
1707940 has the 1707940 v
1707941 has the 1707941 v
1921566 has the 1921566 l
1921567 has the 1921567 l


I search until 2000000

My contributions to Prime Curios

Here are my contributions to Prime curios


11  : The smallest prime which when sandwiched between a two-digit repdigit gives a multiple of 11. In other words 1111, 2112, 3113, 4114, 5115, 6116, 7117, 8118, and 9119 are multiples of 11.


23   : 23 = -2² + 3³


43   : 43 =  4² + 3³


97   : 97 and its double (194) and triple (291) use the same number of characters (five) when expressed in Roman numerals: XCVII, CXCIV, and CCXCI.


109  : The smallest non-trivial prime that is the sum of the reversal of two consecutive primes (109 = R(47) + R(53) = 74 + 35).


239  : 1+3+5+7+....+237+239 = 239+241+243+...+335+337. Note that 239 and 337 are both primes.


251 : The 251st Fibonacci number (F251) has a sum of digits equal to 251. The two smaller prime numbers with this property are 5 and 31


269  : The 269th day of a non-leap year is 26 September (26/9)


617  : 617 = 1!² + 2!² + 3!² + 4!²


991  : 991² = 982081 and 982 + 0 + 8 + 1 = 991.


1009 : The sum of digits of 1009 is a substring of itself and of its square.


1201  1201² = 601+602+603...+1799+1800+1801. With 1201, 601, and 1801 each being prime


1669 : 1669² = 2785561, and 278 * (5/5) * 6 + 1 = 1669


1669 : The smallest prime  that appears in the same position of its own value when the Roman numerals  (from 1 to 3999) are placed in lexicographic order. The other primes with this property are 3623 and 3631


4027  4027⁵ = 3301⁵ + 3169⁵ + 3037⁵ + 2411⁵ + 1481⁵ + 859⁵ + 569⁵. Note that all base numbers and exponents are prime. Found by Takao Nakamura.


4561  : The digits of 4561 (abcd) produce a distinct nine-digit product in the following expression: (a+b+c+d)(ab+cd)(a+bcd)(abc+d)


6833 : 6833² = 46689889, and 4 * 6 + 6898 - 89 = 6833.


8209 : 8209³ = 553185473329, and 5² + 5² + 3² + 1² + 85² + 4² + 7² + 3² + 3² + 29² = 8209.


12637 : The smallest prime such that the differences between the 5 consecutive primes starting with it   are (4,6,6,6): 12637, 12641, 12647, 12653, 12659.


15017 : 15017 = 1!²+2!²+3!²+4!²+5!²


17783 : The smallest prime which is the sum of two, three, four, and five consecutive composite  numbers:
17783 = 8891 + 8892 = 5926 + 5928 + 5929 = 4444 + 4445 + 4446 + 4448 =
3554 + 3555 + 3556 + 3558 + 3560.


28567 : is the smallest prime, which is a Fibonacci number (F(23)prime) and an anagram of a triangular number (67528 = T(367)prime).


41579 : is the only prime p, such that p and p expressed in some base < 10, taken together are   pandigital. 41579 = 63028 in base 9.


38981039 : The smallest number whose square begins and ends with the same seven digits: 38981039² = 1519521401519521.


989450477 : The log730 (989450477) starts out equal to the first dozen digits of pi.


298999999999 : The smallest prime with sum of digits equal to 100.

Primes in arithmetic progression, such one is a permutation of the other

Look at 1487, 4817 and 8147. 
They are three primes with the same digits, one is a permutation of the other, and are in arithmetic progression with a common difference of 3300.  

Another examples:

Common difference, first term, second term and last term

3330    1487     4817     8147
3330     2969    6299     9629
3330     11483    14813    18143
30222    11497    41719    71941
504       12713    13217    13721
4500    12739    17239    21739
4500    12757    17257    21757
4500    12799    17299    21799
33300    14821    48121    81421
16650    14831    31481    48131
32292    14897    47189    79481
33300    18503    51803    85103
33300    18593    51893    85193
15948    19543    35491    51439
450    20161    20611    21061
4950    20353    25303    30253    35203*
4950    20359    25309    30259    35209*
3330    20747    24077    27407
4500    23887    28387    32887
27720    25087    52807    80527
33480    25793    59273    92753
13608    25913    39521    53129
33300    25981    59281    92581
4950    26317    31267    36217
33030    26597    59627    92657
450    28933    29383    29833
33300    29669    62969    96269
3330    31489    34819    38149
8352    31489    39841    48193
30330    32969    63299    93629
4500    34961    39461    43961
4950    35407    40357    45307
4050    35491    39541    43591
17946    35671    53617    71563
14076    37561    51637    65713
4950    49547    54497    59447
450    55603    56053    56503
3330    60373    63703    67033    70363*
4950    60757    65707    70657    75607*
3330    61487    64817    68147
3330    62597    65927    69257
4950    62773    67723    72673    77623*
450    63499    63949    64399
450    67829    68279    68729
9450    68713    78163    87613
2772    71947    74719    77491
5004    73589    78593    83597
450    76717    77167    77617
4950    76819    81769    86719
5238    78941    84179    89417
8910    80191    89101    98011
4950    83987    88937    93887    98837 (all primes)
4950    88937    93887    98837
4500    89387    93887    98387
450    92381    92831    93281

*the last term is not prime