#61 2018-10-26 23:12:18

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Re: Massive coincidences!

22.10.2018 I had next coincidence with V. I was in the same shop as I was 19.10.2014 and I thought about V, some moments later (if I remember it well) I heard advertisement (or something similar) about two Vs. Big W encounter and V in it are associated with a film which I at least partly watched in adolescence, it was one of quite few films which I watched in TV.

Today I typed phrase "21 42" 84 in Google and second result was about one of my topics from a Polish forum. In addition, in fifth link there was the word "Autism".

I also saw results of image finding in Google for the word cats, second image in results has the appearance of a black cat and was a link to the page entitled "Nikola Tesla's Cat and Other Feline Fascinations". Black cat, Nikola Tesla, fascination - it is coincident for me. A person with nick fasci-nation in Polish AS forum wrote a comment on own blog (now not available) in which that person concluded that Tesla would be diagnosed with AS or OCD. I was diagnosed with AS and OCD and had coincidences associated with Tesla and fasci-nation and black cat. Tesla could be supposed to have schizotypal personality or schizophrenia, I had diagnoses of schizotypal disorder or schizophrenia in my life.

Coincidence about three arithmetical sequences with sum of numbers 39 and sum of digits 12 is quite large. Sequence: 6, 13, 20 has interesting property: sum of digits in two-digit numbers (13 and 20) is 6 (and 6 is only one-digit member in that sequence), in addition, digits of numbers 13 and 20 sorted ascendingly or descendingly will form four-membered arithmetical sequence: 0, 1, 2, 3.

There is also another interesting phenomenon associated with three sequences: 11, 13, 15; 12, 13, 14; 6, 13, 20 - sum of single digits which at least once occur in numbers forming that sequence is 21! And there are 7 such a digits which, more interestingly, form seven-membered arithmetical sequence! 0, 1, 2, 3, 4, 5, 6.

#62 2018-10-27 23:53:10

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Re: Massive coincidences!

Yesterday and today I saw words Nikola Tesla's Cat and Other Feline Fascinati in Google. Today it was below third picture in results for phrase cats in Google. It was first picture with BLACK cat in search results.

https://r.hswstatic.com/w_907/gif/tesla-cat.jpg
A black cat, like this one, inspired Tesla's fascination with electricity. OLIVER MOGGRIDGE/EYEEM/GETTY IMAGES

Tesla, fascination, black cat!

There is triad of coincident sequences:
- 12, 13, 14
- 11, 13, 15
- 6, 13, 20.
There are 3 arithmetical sequences. In these sequences are together 9 numbers. Sum of numbers in any of these sequences is 39 and sum of digits in these three sequences is 12 (3+9). Number 6 is a "misfit" among nine numbers mentioned earlier because it is one-digit number, not two-digit. The number of digits in two of these three sequences is 6 and in one of them there is 5 digits due to one-digit "misfit" - 6. The number of digits in these three sequences together is 17, not 18 (3+6+9) due to one-digit number 6.

Most coincident with Tesla's 3, 6, 9 date was 18.12.2015. Last two digits form numbers which sorted ascendingly form an arithmetic sequence - 12, 15, 18. Sums of digits in these 3 numbers are: 3, 6, 9!

#63 2018-10-28 19:47:25

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Re: Massive coincidences!

Three coincident dates are associated with the cementery:
- 6.12.2014,
- 13.12.2014,
- 20.12.2014.

In above three dates there are together 7 digits 2 and 7 digits 1.
Sum of digits 2 in these three dates together is 14.
Sum of digits 1 in these three dates together is 7.
Sum of 14 digits mentioned above is 21.
The sum of digits 2 is two times larger than sum of digits 1. It was in 2014 year.
Number of digits 7 in the dates mentioned above is 0. It was in 2014 year.

Dates:
- 7.12.2014,
- 14.12.2014,
- 21.12.2014
do not give such a coincidence. Number of digits 1 in them together is 8, not 7 and sum of digits 2 and 1 together in these three dates together is 22, not 21; number of digits 2 and 1 together there is not 14 (((21+7):2) or (21-7)), but 15.

Sum of numbers 6, 13, 20 is 39 and sum of that numbers read backwards is also 39 (02+31+6, 02 is counted as 2, not as 0,2), although the sequence 6, 13, 20 is not palindromic or composed only by one-digit numbers.

#64 2018-10-29 23:16:59

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Re: Massive coincidences!

Today I looked at last comments in certain dictionary. On the top of the page I found word (let's name iit U) which has the same first two letters as Polish name of V of some of my previous posts. In addition, fourth, fifth, sixth and seventh letters of U were the same as third, fourth, fifth and sixth letters of Polish name of V! Word U has nine letters, word V has seven letters.
In addition, on the page with recent comments in that dictionary I saw comment in entry "V" (literally, the comment was in entry named just V). Comment for entry U was written about 9 p.m. 29.10.2018, comment for entry V (literally V) was written about 4 p.m. 29.10.2018. There were 10 comments between the two mentioned earlier.

From the digits in date 6.12.2014 a geometrical sequence with sum of numbers 21 can be formed: 1, 4, 16.
From the digits in date 6.12.2014 a geometrical sequence with sum of numbers 42 can be formed: 6, 12, 24.
Two arithmetical sequences with sum of numbers 42 can be formed from digits in 6.12.2014:
- 2, 14, 26;
- 6, 14, 22.

Constant arithmetic or geometric sequence 12, 12, 12 (formed by numbers meaning month in dates: 6.12.2014, 13.12.2014, 20.12.2014) has peculiar properties associated with 3, 6, 9:
1. There are 3 numbers in the sequence 12, 12, 12.
2. There are 6 digits in that sequence.
3. Sum of digits in that sequence is 9.
4. Sum of first digits in numbers forming sequence 12, 12, 12 is 3 (1+1+1).
5. Sum of second digits in numbers forming that sequence is 6 (2+2+2).
6. Sum of numbers in that sequence is 36.
36 is the number constructed from digits 3 and 6.
Sum of digits in 36 is 9.
36 = 6*6 = (3+6+9)+(3+6+9)
7. Product of first digits of numbers 12, 12, 12 is 1 (1*1*1) and product of second digits of these numbers is 8 (2*2*2).
When we write 1 and 8 next to each other, we will receive 18.
18 is 3+6+9.

In addition, quotients of numbers of 12, 12, 12 sequence and sums of digits forming that sequence form constant arithmetical and geometrical sequence with sum 12 and term (and arithmetical mean) 4:
12:(2+1) = 12:3 = 4
12:(2+1) = 12:3 = 4
12:(2+1) = 12:3 = 4
12.4 is a pair of numbers with somewhat interesting properties:
1. Numbers of 12.4 sorted ascendingly or descendingly form a geometric sequence with sum 7: 1, 2, 4.
2. From the phrase 12.4 number 21 can be formed (and also number 12).
3. From the phrase 12.4 number 42 can be formed (and also number 24).
4. From the digits of numbers 12 and 4 numbers 21 and 4 can be formed, product of 21 and 4 is 84.
5. (1+2)*4 = 3*4 = 12
6. (1-2)*4 = (-1)*4 = (-4) [absolute value of -4 is 4]
7. (1*2)*4 = 2*4 = 8 [product of digits forming phrase 12.4 is the same as arithmetic mean of numbers 12 and 4]

#65 2018-10-30 23:39:18

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Re: Massive coincidences!

Today about 6:30 p.m. next comment in entry V in the dictionary mentioned in my previous post was written.

We have the constant sequence: 12, 12, 12.
Sum of digits in one of members in that sequence is 3.
Arithmetic mean of members of that sequence is, of course, 3.
Sum of products of first and second digits of numbers in 12, 12, 12 sequence is the same as sum of the digits in that sequence:
1*1*1 = 1
2*2*2 = 8
1+8 = 9
1+2+1+2+1+2 = 3+3+3 = 9
18 = 3+6+9

Let's look at the sequence 84, 42, 21. Let's divide numbers in that sequence by differences of digits in its terms:
84:(8-4) = 84:4 = 21
42:(4-2) = 42:2 =21
21:(2-1) = 21:1 = 21
We received constant geometrical or arimetrical sequence - 21, 21, 21.
Sum of first digits in the sequence 21, 21, 21 is 6 (2+2+2).
Sum of second digits in the sequence 21, 21, 21 is 3 (1+1+1).
Let's write sums of first and second digits next to each other: we receive number 63. Sum of digits in that number is the same as sum of numbers in 21, 21, 21 and it is 9.
Difference between arithmetical mean of 21, 21, 21 and arithmetical mean of 12, 12, 12 is 9.
Difference between the sum of numebrs in 21, 21, 21 and the sum of numbers in 12, 12, 12 is 27 (63-36). 27 is 3*3*3 = 3*9.
Sum of products of first and second digits of numbers in 21, 21, 21 sequence is the same as sum of the digits in that sequence:
2*2*2 = 8
1*1*1 = 1
8+1 = 9
2+1+2+1+2+1 = 3+3+3 = 9
81 = 9*9 = 3*3*3*3
Sequence 21, 21, 21 has 3 numbers and 6 digits.

#66 2018-10-31 23:38:24

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Re: Massive coincidences!

Today I typed "11 24 37" "11 42 73" in Google. There were 7 results.
I typed also "11 24 37" "11 42 73" "17 24 31" "13 42 71" and found 2 results.
Every of four arithmetical sequences mentioned earlier has 3 terms and 6 digits together in these terms. Sums of digits in these sequences are 18 (3+6+9).
Sums of numbers in these sequences are 126 (21*6) or 72 ((3+6+9)*4).
These sequences written backwards form another arithmetical sequences (one of three different from original one).

I wrote about coincidences associated with these four arithmetical sequences 31.08.2018 here: http://www.nld.pun.pl/viewtopic.php?pid=677#p677 Here is written about prime numbers which are fascinately associated with these four sequences.

In Google I found 2 results for the phrase "All extreme members of these four" which were associated with my posts.

I am interested if anyone else in history noticed that prime numbers-associated pattern in sequences: 11, 24, 37; 11, 42, 73; 17, 24, 31; 13, 42, 71.
Extreme members of these sequences are all prime numbers (11, 37, 73, 17, 31, 13, 71). In these prime numbers only three (21:7 = 3) sort of digits are present, they form a product which is 21 (1*3*7 = 21).
Common differences of four arithmetical sequences mentioned earlier are all prime numbers (13, 31, 7, 29).
It is interesting if there is any other group of arithmetical or geometrical sequences which has such a property.

It is not the end of prime number-associated phenomena. When we sort digits forming any of these four arithmetical sequences ascendingly and then form two-digit numbers, we will receive 11, 23, 47 - three prime numbers which sums of digits are another prime numbers. We can wrote next numbers in that sequence (which is not arithmetic or geometric), these numbers are 83, 131, 191, 263. They are also additive prime numbers. Amazing!

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#67 2018-11-01 22:32:06

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Re: Massive coincidences!

11, 42, 73; 13, 42, 71 and 33, 42, 51 are somewhat similar arithmetical sequences:
- all have 3 terms,
- all have 6 digits,
- all have the same sum of numbers (21*6 = 126, sum of digits in 126 is 9),
- all have the same sum of digits (18, 18 = 3+6+9, sum of digits in 18 is 9),
- all have the same sum of numbers in arithmetical sequences formed by reading "original" sequences backwards [37, 24, 11; 17, 24, 31 and 15, 24, 33 - sum of numbers in these sequences is 72, which is multiple of 18 (3+6+9), sum of digits in 72 is 9],
- first digits in these sequences form arithmetical sequences (1, 4, 7; 1, 4, 7; 3, 4, 5 - two of these triplets form number 147 (21*7) when written next to each other, sums of these sequences are 12),
- second digits in these sequences form arithmetical sequences (1, 2, 3; 3, 2, 1; 3, 2, 1 - sums of these triplets are 6).

#68 2018-11-02 23:21:44

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Re: Massive coincidences!

11, 23, 47, 83, 131, 191, 263 - it is not a geometrical or arithmetical sequence, but it has interesting properties - sums of these numbers are additive prime numbers! These sums are: 2, 5, 11, 11, 5, 11, 11. Sum of digits in number 11 is 2.

11, 23, 47, 83, 131, 191, 263. It is probably the smallest such a sequence of additive prime numbers. Other as long sequences of additive primes appear to be hard to find or maybe even do not exist.

Numbers 2, 5, 11, 23, 47 are five first Thabit numbers.

Here are another sequences of prime numbers: http://mathforum.org/kb/message.jspa?messageID=415186.
A sequence 41, 43, 47, 53, 61, 71, 83, 97, 113, 131, 151, 173, 197, 223, 251, 281, 313, 347, 383, 421, 461, 503, 547, 593, 641, 691, 743, 797, 853, 911, 971, 1033, 1097, 1163, 1231, 1301, 1373, 1447, 1523, 1601 is mentioned here. There are 7 consecutive additive primes in that sequence (bolded), but not all seven have digit sum which is a prime number.
A sequence 17, 29, 53, 89, 137, 197, 269, 353, 449, 557, 677, 809, 953, 1109, 1277 also has seven consecutive additive prime numbers. However, not all these primes (bolded) are primes with additive primes as the sum of their digits.

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#69 2018-11-04 20:05:33

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Re: Massive coincidences!

From digits of the ascending arithmetical sequences:
- 11, 24, 37;
- 11, 42, 73;
- 13, 42, 71;
- 17, 24, 31;
phrase 21 7 can be formed.

Absolute values of common differences of sequences formed by reading these four sequences backwards are prime numbers.

It is interesting if someone else noticed pattern associated with prime numbers and digits: 1, 1, 2, 3, 4, 7. From these digits four ascending arithmetical sequences can be formed.
These sequences have middle terms and arithmetical means 42 or 24, sum of digits in these two numbers is 6 and arithmetical mean of digits is 3.
Sums of these three-membered arithmetical sequences are 72 and 126 and 72 and 126 are divisible by 18 (3+6+9).
Sum of digits in these sequences is 18 (3+6+9).

Products of digits in extreme terms of four arithmetical sequences mentioned above are 21:
- 1*1*3*7 = 21
- 1*1*7*3 = 21
- 1*3*7*1 = 21
- 1*7*3*1 = 21.

What is very interesting, two arithmetical sequences with three (21u]7[/u]) numbers and sum 21 and arithmetical mean 7 (and middle term 7 also) can be formed from digits in extreme members of four ASes mentioned above:
- 1, 7, 13;
- 3, 7, 11.
In addition, these two ASes have the same products of their four digits as sums of three numbers which form them!
1*7*1*3 = 1+7+13 = 21
3*7*1*1 = 3+7+11 = 21
These arithmetical sequences (with the property mentioned above) are probably really rare. Another two which give such a property are 0, 0, 0 and 1, 2, 3.
0*0*0  = 0+0+0 = 0
1+2+3 = 1*2*3 = 6

The sequence 3, 7, 11 is especially interesting. It is formed by three additive prime numbers (3, 7, 11, sum of digits in 11 is 2 and it is a prime) and gives its descending counterpart (11, 7, 3) when read backwards. Another (even more interesting, because its CD is also an additive prime number - 2) sequence which has such a property is 3, 5, 7. Number 357 is the result of multiplication of 21 and 17. Numbers 21 and 17 can be formed from digits in the sequence 11, 24, 37.

First five Thabit numbers are additive prime numbers associated with digits 1, 1, 2, 3, 4, 7.
11, 23, 47 - third, fourth and fifth thabit numbers. First is 2, second is 5.
2, 5, 11 - three first Thabit numbers and sum of digits in 11, 23, 47!
Sum of 1st, 2nd and 3rd Thabit numbers is 18 and sum of digits in these three numbers is 18:2 = 9 (2 is the smallest Thabit number and sum of digits in the smallest term of 11, 23, 47 sequence).
Sum of 11, 23, 47 is 81, which is 18 read backwards. In addition, 81 is 9*9 and 18 is 2*9. Product of two of first factors in 2*9 and 9*9 is 2*9 = 18.
Sum of 18 and 81 is 99.

#70 2018-11-04 20:07:52

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Re: Massive coincidences!

Sorry, there is some sort of error in my previous post...

What is very interesting, two arithmetical sequences with three (3 is 21:7) numbers and sum 21 and arithmetical mean 7 (and middle term 7 also) can be formed from digits in extreme members of four ASes mentioned above:
- 1, 7, 13;
- 3, 7, 11.
In addition, these two ASes have the same products of their four digits as sums of three numbers which form them!
1*7*1*3 = 1+7+13 = 21
3*7*1*1 = 3+7+11 = 21
These arithmetical sequences (with the property mentioned above) are probably really rare. Another two which give such a property are 0, 0, 0 and 1, 2, 3.
0*0*0  = 0+0+0 = 0
1+2+3 = 1*2*3 = 6

#71 2018-11-05 23:42:57

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Re: Massive coincidences!

Thabit numbers are also called "321 numbers". Phrase "321" can be formed from digits in numbers 11, 24, 37.
All sums of digits in members of 11, 23, 47, 83, 131, 191, 263 are prime Thabit numbers 2, 5 or 11. Sums of digits in these sums are 2 or 5 - two first Thabit primes.

11, 24, 37. 11, 42, 73. 13, 42, 71. 17, 24, 31. Sum of these four sequences is 396. Number formed by digits: 3, 6, 9! It is divisible by 3+6+9 (18) - 369 is not divisible by 18 because it is an odd number and 18 is an even number.
396: 18 = 22 (that two-digit number has the same sum of digits as product of them).
Arithmetical mean of sums of numbers in four sequences mentioned earlier is 99 (72+126+72+126):4 = (198+198):2 = 99.
Arithmetical mean of numbers in these four sequences is 396:12 = 33.
33 - arithmetical mean of digits in this number is 3.
33 - sum of digits in this number is 6.
33 - product of digits in this number is 9.

#72 2018-11-05 23:43:14

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Re: Massive coincidences!

From combinations of digits of "miraculous sequences" 84, 42, 21 and 21, 42, 84 geometrical sequences with sums: 21, 42, 84, 63 can be formed.
1. 2+1 = 3
4+2 = 6
8+4 = 12
3, 6, 12 - geometrical sequence with sum 21!
2. (2*1)2:1) = 2:2 = 1
(4*2)4:2) = 8:2 = 4
(8*4)8:4) = 32:2 = 16
1, 4, 16 - geometrical sequence with sum 21!
3. 21:(2+1) = 21:3 = 7
42:(4+2) = 42:6 = 7
84:(8+4) = 84:12 = 7
7, 7, 7 - geometrical sequence with sum 21!
4. 2*1 = 2
4*2 = 8
8*4 = 32
2, 8, 32 - geometrical sequence with sum 42!
5. (2+1)*(2:1) = 3*2 = 6
(4+2)*(4:2) = 6*2 = 12
(8+4)*(8:4) = 12*2 = 24
6, 12, 24 - geometrical sequence with sum 42!
6. 21 read backwards - 12
42 read backwards - 24
84 read backwards - 48
12, 24, 48 - geometrical sequence with sum 84!
7. (2*1)*(2:1) = 2*2 = 4
(4*2)*(4:2) = 8*2 = 16
(8*4)*(8:4) = 32*2 = 64
4, 16, 64 - geometrical sequence with sum 84!
8. 21:(2-1) = 21:1 = 21
42:(4-2) = 42:2 = 21
84:(8-4) = 84:4 = 21
21, 21, 21 - geometrical sequence with sum 63 (21*3) and arithmetical mean 21!
9. (2+1)*(2-1) = 3*1 = 3
(4+2)*(4-2) = 6*2 = 12
(8+4)*(8-4) = 12*4 = 48
3, 12, 48 - geometrical sequence with sum 63 (21*3) and arithmetical mean 21!

#73 2018-11-05 23:44:42

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Re: Massive coincidences!

The same content again... Emoticons instead of ":"...

From combinations of digits of "miraculous sequences" 84, 42, 21 and 21, 42, 84 geometrical sequences with sums: 21, 42, 84, 63 can be formed.
1. 2+1 = 3
4+2 = 6
8+4 = 12
3, 6, 12 - geometrical sequence with sum 21!
2. (2*1) : (2:1) = 2:2 = 1
(4*2) : (4:2) = 8:2 = 4
(8*4) : (8:4) = 32:2 = 16
1, 4, 16 - geometrical sequence with sum 21!
3. 21:(2+1) = 21:3 = 7
42:(4+2) = 42:6 = 7
84:(8+4) = 84:12 = 7
7, 7, 7 - geometrical sequence with sum 21!
4. 2*1 = 2
4*2 = 8
8*4 = 32
2, 8, 32 - geometrical sequence with sum 42!
5. (2+1)*(2:1) = 3*2 = 6
(4+2)*(4:2) = 6*2 = 12
(8+4)*(8:4) = 12*2 = 24
6, 12, 24 - geometrical sequence with sum 42!
6. 21 read backwards - 12
42 read backwards - 24
84 read backwards - 48
12, 24, 48 - geometrical sequence with sum 84!
7. (2*1)*(2:1) = 2*2 = 4
(4*2)*(4:2) = 8*2 = 16
(8*4)*(8:4) = 32*2 = 64
4, 16, 64 - geometrical sequence with sum 84!
8. 21:(2-1) = 21:1 = 21
42:(4-2) = 42:2 = 21
84:(8-4) = 84:4 = 21
21, 21, 21 - geometrical sequence with sum 63 (21*3) and arithmetical mean 21!
9. (2+1)*(2-1) = 3*1 = 3
(4+2)*(4-2) = 6*2 = 12
(8+4)*(8-4) = 12*4 = 48
3, 12, 48 - geometrical sequence with sum 63 (21*3) and arithmetical mean 21!

#74 2018-11-11 23:09:34

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Re: Massive coincidences!

Difference between middle and the smallest term of sequence 21, 42, 84 is 21 (42-21). Sum of digits in number 21 is 3.
Difference between the largest and midlle term of that sequence is 42 (84-42). Sum of digits in number 42 is 6.
Difference between the largest and the smallest term of that sequence is 63 (84-21). Sum of digits in number 63 is 9. In addition, two digits forming that number are 3 and 6.

First three numbers of 11, 23, 47, 83, 131, 191, 263 sequence are numbers which can be formed by substracting 1 from the numbers forming geometrical sequence 12, 24, 48.

First three Thabit numbers (2, 5, 11) are numbers which can be formed by subtracting 1 from numbers which are sums of digits in numbers 21, 42, 84. These sum of digits are: 3, 6, 12.

Second, third and fourth Thabit numbers (5, 11, 23) are formed by subtracting 1 from numbers forming the geometrical sequence 6, 12, 24. Sum of these three Thabit numbers is 39.

First, third and fifth Thabit numbers (2, 11, 47) are formed by subtracting 1 from digits forming the sequence 3, 12, 48.

#75 2018-11-13 23:34:29

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Re: Massive coincidences!

Some time ago I read about numbers 7, 11, 13 associated with certain text. It could be in summer or autumn 2017, relatively shortly after ending my third going to day hospital, which started 27.4.2017 and ended 24.7.2017. There is coincidence between numbers 7, 11, 13 and 27, 4 or 24, 7!

It is pretty large coincidence which I noticed maybe only today.
Sum of numbers 7, 11, 13 is the same as sum of numbers 27 and 4 or 24 and 7 and it is 31, a prime number.
Sum of digits in numbers 7, 11, 13 is the same as sum of digits in numbers 27 and 4 or 24 and 7 and it is 13, a prime number which is formed by reading digits in number 31 backwards!

In addition, sums of digits in numbers 7, 11 and 13 form the same digits as digits forming numbers 27 and 4 or 24 and 7!
7 = 7
1+1 = 2
1+3 = 4

Product of digits in numbers 7, 11, 13 is 21 (7*1*1*1*3). It is smaller than sum of these three numbers: 7+11+13 = 31. From the digits of numbers 7, 11, 13 two extreme members of arithmetical sequences: 11, 24, 37; 11, 42, 73; 13, 42, 71; 17, 24, 31 can be formed.

From digits of numbers 7, 11, 13 two ascending arithmetical sequences with sum of digits 21 and arithmetical mean 7 can be formed: 1, 7, 13 and 3, 7, 11.

I also found interesting triplet of additive prime numbers: 7, 29, 137. These are three additive prime numbers.
1. Sum of these three numbers is 173, an additive prime number which is formed by the same digits as the largest numbers in triplet 7, 29, 137.
2. Sums of digits in numbers 7, 29, 137 are all additive prime numbers:
7 = 7
2+9 = 11
1+3+7 = 11
3. Sum of sums of digits in 7, 29, 137 sequence is also an additive prime number!
7+11+11 = 29, sum of digits in 29 - 11, 11 occurs twice as a sum of digits in numbers 7, 29, 139; 29 is a member of "original" sequence.
4. Sum of digits in sums of digits in numbers of 7, 29, 137 sequence is also an additive prime number!
7+(1+1)+(1+1) = 7+2+2 = 11

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