This site is supported by donations to The OEIS Foundation.

# User:Karsten Meyer

I exist. I went to school. I have the "Mittlere Reife" and "Fachhochschulreife". My Profession is Chemical laboratory technician. I went for some Years to the "Fachhochschule Gießen-Friedberg" in Germany. I have written some Articles in "de.wikipedia.org", "de.wikibooks.org" and "en.citizendium.org".

I am interested in Number theory, especially Prime numbers. Pseudoprimes, Lucas sequences and others.

--

Sqequences, where i am participated in:
• OEIS: A111305 - Composite numbers n such that a^(n-1) = 1 mod n only when a = 1 mod n.
• OEIS: A104189 - Prime numbers arising from Schorn's proof that there are infinitely many primes.
• OEIS: A112540 - Numbers n such that (15n-4; 15n-2; 15n+2; 15n+4) is a prime quadruplet.
• OEIS: A051015 - Zeisel numbers.

## 1729

1729 is my favorite number because it is not only a Carmichael number. 1729 is the smallest Carmichael number you can construct with the formula (6n + 1) * (12 + 1) * (18 + 1). 1729 is the smallest absolute euler pseudoprime. 1729 is the product of 7 * 13. 1729 is also a Zeisel number.

## New Sequence

$\frac{1}{1} \, \frac{4}{3} \, \frac{9}{6} \, \frac{16}{10} \, \frac{25}{15} \, \frac{36}{21} \, \frac{49}{28} \, \frac{64}{36} \, \frac{81}{45} \, \frac{100}{55} \, \frac{121}{66} \, \frac{144}{78} \, \frac{169}{91} \, \frac{196}{105} \, ...$
$\sum_{i=1}^n{\frac{1}{\Delta i}} = \frac{n^2}{\Delta n}$
$\lim_{n \to \infty} \frac{n^2}{\Delta n} = 2$

## Distribution Primenumbers and fermat Pseudoprimes

 Number n Number of primes p Number of fermat pseudoprimes q 10 4 0 100 25 27 1000 168 429 10.000 1229 5099

## Fingerprint of natural numbers

### Natural numbers

Natural numbers have a common pattern:

 1 2 3 4 5 6 1: 1 1 1 1 1 1 X: X X X X X X X: X X X X X X X: X X X X X X X: X X X X X X A: A 1 A 1 A 1

In der obersten Zeile befinden sich immer Einsen und in der untersten Zeile befinden sich immer im Wechsel A und 1, wobei A für $n-1\$ steht. Dies ist also keine Charakteristik, die für Primzahlen typisch ist.

 Color Formular $a^{\lambda(n)}\$ $a^{\frac{\lambda(n)}{2}}$

A prime number has a typical pattern:

 1 2 3 4 5 6 1: 1 1 1 1 1 1 X: X X 1 X X 1 X: X X A X X 1 X: X X 1 X X 1 X: X X A X X 1 A: A 1 A 1 A 1

### Numbers

 1 1: 1
 1 2 1: 1 1 2: 2 1
 1 2 1: 1 1 2: 2 4 3: 3 1
 1 2 3 4 1: 1 1 1 1 2: 2 4 3 1 3: 3 4 2 1 4: 4 1 4 1
 1 2 1: 1 1 2: 2 4 3: 3 3 4: 4 4 5: 5 1
 1 2 3 4 5 6 1: 1 1 1 1 1 1 2: 2 4 1 2 4 1 3: 3 2 6 4 5 1 4: 4 2 1 4 2 1 5: 5 4 6 2 3 1 6: 6 1 6 1 6 1
 1 2 1: 1 1 2: 2 4 3: 3 1 4: 4 0 5: 5 1 6: 6 4 7: 7 1
 1 2 3 4 5 6 1: 1 1 1 1 1 1 2: 2 4 8 7 5 1 3: 3 0 0 0 0 0 4: 4 7 1 4 7 1 5: 5 7 8 4 2 1 6: 6 0 0 0 0 0 7: 7 4 1 7 4 1 8: 8 1 8 1 8 1
 1 2 3 4 1: 1 1 1 1 2: 2 4 8 6 3: 3 9 7 1 4: 4 6 4 6 5: 5 5 5 5 6: 6 6 6 6 7: 7 9 3 1 8: 8 4 2 6 9: 9 1 9 1
2 3 4 5 6 7 8 9 10
 1 2 3 4 5 6 7 8 9 10 1: 1 1 1 1 1 1 1 1 1 1 2: 2 4 8 5 10 9 7 3 6 1 3: 3 9 5 4 1 3 9 5 4 1 4: 4 5 9 3 1 4 5 9 3 1 5: 5 3 4 9 1 5 3 4 9 1 6: 6 3 7 9 10 5 8 4 2 1 7: 7 5 2 3 10 4 6 9 8 1 8: 8 9 6 4 10 3 2 5 7 1 9: 9 4 3 5 1 9 4 3 5 1 10: 10 1 10 1 10 1 10 1 10 1
 1 2 1: 1 1 2: 2 4 3: 3 9 4: 4 4 5: 5 1 6: 6 0 7: 7 1 8: 8 4 9: 9 3 10: 10 4 11: 11 1
 1 2 3 4 5 6 7 8 9 10 11 12 1: 1 1 1 1 1 1 1 1 1 1 1 1 2: 2 4 8 3 6 12 11 9 5 10 7 1 3: 3 9 1 3 9 1 3 9 1 3 9 1 4: 4 3 12 9 10 1 4 3 12 9 10 1 5: 5 12 8 1 5 12 8 1 5 12 8 1 6: 6 10 8 9 2 12 7 3 5 4 11 1 7: 7 10 5 9 11 12 6 3 8 4 2 1 8: 8 12 5 1 8 12 5 1 8 12 5 1 9: 9 3 1 9 3 1 9 3 1 9 3 1 10: 10 9 12 3 4 1 10 9 12 3 4 1 11: 11 4 5 3 7 12 2 9 8 10 6 1 12: 12 1 12 1 12 1 12 1 12 1 12 1
 1 2 3 4 5 6 1: 1 1 1 1 1 1 2: 2 4 8 2 4 8 3: 3 9 13 11 5 1 4: 4 2 8 4 2 8 5: 5 11 13 9 3 1 6: 6 8 6 8 6 8 7: 7 7 7 7 7 7 8: 8 8 8 8 8 8 9: 9 11 1 9 11 1 10: 10 2 6 4 12 8 11: 11 9 1 11 9 1 12: 12 4 6 2 10 8 13: 13 1 13 1 13 1
 1 2 3 4 1: 1 1 1 1 2: 2 4 8 1 3: 3 9 2 6 4: 4 1 4 1 5: 5 10 5 10 6: 6 6 6 6 7: 7 4 13 1 8: 8 4 2 1 9: 9 6 9 6 10: 10 10 10 10 11: 11 1 11 1 12: 12 9 3 6 13: 13 4 7 1 14: 14 1 14 1
 1 2 3 4 1: 1 1 1 1 2: 2 4 8 0 3: 3 9 11 1 4: 4 0 0 0 5: 5 9 13 1 6: 6 4 8 0 7: 7 1 7 1 8: 8 0 0 0 9: 9 1 9 1 10: 10 4 8 0 11: 11 9 3 1 12: 12 2 8 0 13: 13 9 5 1 14: 14 4 8 0 15: 15 1 15 1
11 12 13 14 15 16

Siebzehn:

Achtzehn:

Neunzehn:

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2: 2 4 8 16 13 7 14 9 18 17 15 11 3 6 12 5 10 1 3: 3 9 8 5 15 7 2 6 18 16 10 11 14 4 12 17 13 1 4: 4 16 7 9 17 11 6 5 1 4 16 7 9 17 11 6 5 1 5: 5 6 11 17 9 7 16 4 1 5 6 11 17 9 7 16 4 1 6: 6 17 7 4 5 11 9 16 1 6 17 7 4 5 11 9 16 1 7: 7 11 1 7 11 1 7 11 1 7 11 1 7 11 1 7 11 1 8: 8 7 18 11 12 1 8 7 18 11 12 1 8 7 18 11 12 1 9: 9 5 7 6 16 11 4 17 1 9 5 7 6 16 11 4 17 1 10: 10 5 12 6 3 11 15 17 18 9 14 7 13 16 8 4 2 1 11: 11 7 1 11 7 1 11 7 1 11 7 1 11 7 1 11 7 1 12: 12 11 18 7 8 1 12 11 18 7 8 1 12 11 18 7 8 1 13: 13 17 12 4 14 11 10 16 18 6 2 7 15 5 8 9 3 1 14: 14 6 8 17 10 7 3 4 18 5 13 11 2 9 12 16 15 1 15: 15 16 12 9 2 11 13 5 18 4 3 7 10 17 8 6 14 1 16: 16 9 11 5 4 7 17 6 1 16 9 11 5 4 7 17 6 1 17: 17 4 11 16 6 7 5 9 1 17 4 11 16 6 7 5 9 1 18: 18 1 18 1 18 1 18 1 18 1 18 1 18 1 18 1 18 1

20:

21:

 1 2 3 4 5 6 1: 1 1 1 1 1 1 2: 2 4 8 16 11 1 3: 3 9 6 18 12 15 4: 4 16 1 4 16 1 5: 5 4 20 16 17 1 6: 6 15 6 15 6 15 7: 7 7 7 7 7 7 8: 8 1 8 1 8 1 9: 9 18 15 9 18 15 10: 10 16 13 4 19 1 11: 11 16 8 4 2 1 12: 12 18 6 9 3 15 13: 13 1 13 1 13 1 14: 14 7 14 7 14 7 15: 15 15 15 15 15 15 16: 16 4 1 16 4 1 17: 17 16 20 4 5 1 18: 18 9 15 18 9 15 19: 19 4 13 16 10 1 20: 20 1 20 1 20 1

Fünfundzwanzig:

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2: 2 4 8 16 7 14 3 6 12 24 23 21 17 9 18 11 22 19 13 1 3: 3 9 2 6 18 4 12 11 8 24 22 16 23 19 7 21 13 14 17 1 4: 4 16 14 6 24 21 9 11 19 1 4 16 14 6 24 21 9 11 19 1 5: 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6: 6 11 16 21 1 6 11 16 21 1 6 11 16 21 1 6 11 16 21 1 7: 7 24 8: 8 14 9: 9 6 4 11 10: 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 11: 11 21 12: 12 1 20 1 20 1 13: 13 1 1 1 1 1 14: 14 4 8 16 11 1 15: 15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 16: 16 4 20 16 17 1 17: 17 1 8 1 8 1 18: 18 16 13 4 19 1 19: 19 16 8 4 2 1 20: 20 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 21: 21 4 1 16 4 1 22: 22 16 20 4 5 1 23: 23 4 13 16 10 1 24: 24 1 24 1 24 1 24 1 24 1 24 1 24 1 24 1 24 1 24 1

Achtundzwanzig:

 1 2 3 4 5 6 1: 1 1 1 1 1 1 2: 2 4 8 16 4 8 3: 3 9 27 25 19 1 4: 4 16 8 4 16 8 5: 5 25 13 9 17 1 6: 6 8 20 8 20 8 7: 7 21 7 21 7 21 8: 8 8 8 8 8 8 9: 9 25 1 9 25 1 10: 10 16 20 4 12 8 11: 11 9 15 25 23 1 12: 12 4 20 16 24 8 13: 13 1 13 1 13 1 14: 14 0 0 0 0 0 15: 15 1 15 1 15 1 16: 16 4 8 16 4 8 17: 17 9 13 25 5 1 18: 18 16 8 4 16 8 19: 19 25 27 9 3 1 20: 20 8 20 8 20 8 21: 21 21 21 21 21 21 22: 22 8 8 8 8 8 23: 23 25 15 9 11 1 24: 24 16 20 4 12 8 25: 25 9 1 25 9 1 26: 26 4 20 16 24 8 27: 27 1 27 1 27 1

33:

 1 2 3 4 5 6 7 8 9 10 1: 1 1 1 1 1 1 1 1 1 1 1 2: 2 4 8 16 32 31 29 25 17 1 3: 3 9 27 15 12 3 9 27 15 12 4: 4 16 31 25 1 4 16 31 25 1 5: 5 25 26 31 23 16 14 4 20 1 6: 6 3 18 9 21 27 30 15 24 12 7: 7 16 13 25 10 4 28 31 19 1 8: 8 31 17 4 32 25 2 16 29 1 9: 9 15 3 27 12 9 15 3 27 12 10: 10 1 10 1 10 1 10 1 10 1 11: 11 22 11 22 11 22 11 22 11 22 12: 12 12 12 12 12 12 12 12 12 12 13: 13 4 19 16 10 31 7 25 28 1 14: 14 31 5 4 23 25 20 16 26 1 15: 15 27 9 3 12 15 27 9 3 12 16: 16 25 4 31 1 16 25 4 31 1 17: 17 25 29 31 32 16 8 4 2 1 18: 18 27 24 3 21 15 6 9 30 12 19: 19 31 28 4 10 25 13 16 7 1 20: 20 4 14 16 23 31 26 25 5 1 21: 21 12 21 12 21 12 21 12 21 12 22: 22 22 22 22 22 22 22 22 22 22 23: 23 1 23 1 23 1 23 1 23 1 24: 24 15 30 27 21 9 18 3 6 12 25: 25 31 16 4 1 25 31 16 4 1 26: 26 16 20 25 23 4 5 31 14 1 27: 27 3 15 9 12 27 3 15 9 12 28: 28 25 7 31 10 16 19 4 13 1 29: 29 16 2 25 32 4 17 31 8 1 30: 30 9 6 15 21 3 24 27 18 12 31: 31 4 25 16 1 31 4 25 16 1 32: 32 1 32 1 32 1 32 1 32 1 1

35

 1 2 3 4 5 6 7 8 9 10 11 12 1: 1 1 1 1 1 1 1 1 1 1 1 1 2: 2 4 8 16 32 29 23 11 22 9 18 1 3: 3 9 27 11 33 29 17 16 13 4 12 1 4: 4 16 29 11 9 1 4 16 29 11 9 1 5: 5 25 20 30 10 15 5 25 20 30 10 15 6: 6 1 6 1 6 1 6 1 6 1 6 1 7: 7 14 28 21 7 14 28 21 7 14 28 21 8: 8 29 22 1 8 29 22 1 8 29 22 1 9: 9 11 29 16 4 1 9 11 29 16 4 1 10: 10 30 20 25 5 15 10 30 20 25 5 15 11: 11 16 1 11 16 1 11 16 1 11 16 1 12: 12 4 13 16 17 29 33 11 27 9 3 1 13: 13 29 27 1 13 29 27 1 13 29 27 1 14: 14 21 14 21 14 21 14 21 14 21 14 21 15: 15 15 15 15 15 15 15 15 15 15 15 15 16: 16 11 1 16 11 1 16 11 1 16 11 1 17: 17 9 13 11 12 29 3 16 27 4 33 1 18: 18 9 22 11 23 29 32 16 8 4 2 1 19: 19 11 34 16 24 1 19 11 34 16 24 1 20: 20 15 20 15 20 15 20 15 20 15 20 15 21: 21 21 21 21 21 21 21 21 21 21 21 21 22: 22 29 8 1 22 29 8 1 22 29 8 1 23: 23 4 22 16 18 29 2 11 8 9 32 1 24: 24 16 34 11 19 1 24 16 34 11 19 1 25: 25 30 15 25 30 15 25 30 15 25 30 15 26: 26 11 6 16 31 1 26 11 6 16 31 1 27: 27 29 13 1 27 29 13 1 27 29 13 1 28: 28 14 7 21 28 14 7 21 28 14 7 21 29: 29 1 29 1 29 1 29 1 29 1 29 1 30: 30 25 15 30 25 15 30 25 15 30 25 15 31: 31 16 6 11 26 1 31 16 6 11 26 1 32: 32 9 8 11 2 29 18 16 22 4 23 1 33: 33 4 27 16 3 29 12 11 13 9 17 1 34: 34 1 34 1 34 1 34 1 34 1 34 1

39

 1 2 3 4 5 6 7 8 9 10 11 12 1: 1 1 1 1 1 1 1 1 1 1 1 1 2: 2 4 8 16 32 25 11 22 5 10 20 1 3: 3 9 27 3 9 27 3 9 27 3 9 27 4: 4 16 25 22 10 1 4 16 25 22 10 1 5: 5 25 8 1 5 25 8 1 5 25 8 1 6: 6 36 21 9 15 12 33 3 18 30 24 27 7: 7 10 31 22 37 25 19 16 34 4 28 1 8: 8 25 5 1 8 25 5 1 8 25 5 1 9: 9 3 27 9 3 27 9 3 27 9 3 27 10: 10 22 25 16 4 1 10 22 25 16 4 1 11: 11 4 5 16 20 25 2 22 8 10 32 1 12: 12 27 12 27 12 27 12 27 12 27 12 27 13: 13 13 13 13 13 13 13 13 13 13 13 13 14: 14 1 14 1 14 1 14 1 14 1 14 1 15: 15 30 21 3 6 12 24 9 18 36 33 27 16: 16 22 1 16 22 1 16 22 1 16 22 1 17: 17 16 38 22 23 1 17 16 38 22 23 1 18: 18 12 21 27 18 12 21 27 18 12 21 27 19: 19 10 34 22 28 25 7 16 31 4 37 1 20: 20 10 5 22 11 25 32 16 8 4 2 1 21: 21 12 18 27 21 12 18 27 21 12 18 27 22: 22 16 1 22 16 1 22 16 1 22 16 1 23: 23 22 38 16 17 1 23 22 38 16 17 1 24: 24 30 18 3 33 12 15 9 21 36 6 27 25: 25 1 25 1 25 1 25 1 25 1 25 1 26: 26 13 26 13 26 13 26 13 26 13 26 13 27: 27 27 27 27 27 27 27 27 27 27 27 27 28: 28 4 34 16 19 25 37 22 31 10 7 1 29: 29 22 14 16 35 1 29 22 14 16 35 1 30: 30 3 12 9 36 27 30 3 12 9 36 27 31: 31 25 34 1 31 25 34 1 31 25 34 1 32: 32 10 8 22 2 25 20 16 5 4 11 1 33: 33 36 18 9 24 12 6 3 21 30 15 27 34: 34 25 31 1 34 25 31 1 34 25 31 1 35: 35 16 14 22 29 1 35 16 14 22 29 1 36: 36 9 12 3 30 27 36 9 12 3 30 27 37: 37 4 31 16 7 25 28 22 34 10 19 1 38: 38 1 38 1 38 1 38 1 38 1 38 1

45

 1 2 3 4 5 6 7 8 9 10 11 12 1: 1 1 1 1 1 1 1 1 1 1 1 1 2: 2 4 8 16 32 19 38 31 17 34 23 1 3: 3 9 27 36 18 9 27 36 18 9 27 36 4: 4 16 19 31 34 1 4 16 19 31 34 1 5: 5 25 35 40 20 10 5 25 35 40 20 10 6: 6 36 36 36 36 36 36 36 36 36 36 36 7: 7 4 28 16 22 19 43 31 37 34 13 1 8: 8 19 17 1 8 19 17 1 8 19 17 1 9: 9 36 9 36 9 36 9 36 9 36 9 36 10: 10 10 10 10 10 10 10 10 10 10 10 10 11: 11 31 26 16 41 1 11 31 26 16 41 1 12: 12 9 18 36 27 9 18 36 27 9 18 36 13: 13 34 37 31 43 19 22 16 28 4 7 1 14: 14 16 44 31 29 1 14 16 44 31 29 1 15: 15 0 0 0 0 0 0 0 0 0 0 0 16: 16 31 1 16 31 1 16 31 1 16 31 1 17: 17 19 8 1 17 19 8 1 17 19 8 1 18: 18 9 27 36 18 9 27 36 18 9 27 36 19: 19 1 19 1 19 1 19 1 19 1 19 1 20: 20 40 35 25 5 10 20 40 35 25 5 10 21: 21 36 36 36 36 36 36 36 36 36 36 36 22: 22 34 28 31 7 19 13 16 37 4 43 1 23: 23 34 17 31 38 19 32 16 8 4 2 1 24: 24 36 9 36 9 36 9 36 9 36 9 36 25: 25 40 10 25 40 10 25 40 10 25 40 10 26: 26 1 26 1 26 1 26 1 26 1 26 1 27: 27 9 18 36 27 9 18 36 27 9 18 36 28: 28 19 37 1 28 19 37 1 28 19 37 1 29: 29 31 44 16 14 1 29 31 44 16 14 1 30: 30 0 0 0 0 0 0 0 0 0 0 0 31: 31 16 1 31 16 1 31 16 1 31 16 1 32: 32 34 8 31 2 19 23 16 17 4 38 1 33: 33 9 27 36 18 9 27 36 18 9 27 36 34: 34 31 19 16 4 1 34 31 19 16 4 1 35: 35 10 35 10 35 10 35 10 35 10 35 10 36: 36 36 36 36 36 36 36 36 36 36 36 36 37: 37 19 28 1 37 19 28 1 37 19 28 1 38: 38 4 17 16 23 19 2 31 8 34 32 1 39: 39 36 9 36 9 36 9 36 9 36 9 36 40: 40 25 10 40 25 10 40 25 10 40 25 10 41: 41 16 26 31 11 1 41 16 26 31 11 1 42: 42 9 18 36 27 9 18 36 27 9 18 36 43: 43 4 37 16 13 19 7 31 28 34 22 1 44: 44 1 44 1 44 1 44 1 44 1 44 1

49

51

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2: 2 4 8 16 32 13 26 1 2 4 8 16 32 13 26 1 3: 3 9 27 30 39 15 45 33 48 42 24 21 12 36 6 18 4: 4 16 13 1 4 16 13 1 4 16 13 1 4 16 13 1 5: 5 25 23 13 14 19 44 16 29 43 11 4 20 49 41 1 6: 6 36 12 21 24 42 48 33 45 15 39 30 27 9 3 18 7: 7 49 37 4 28 43 46 16 10 19 31 13 40 25 22 1 8: 8 13 2 16 26 4 32 1 8 13 2 16 26 4 32 1 9: 9 30 15 33 42 21 36 18 9 30 15 33 42 21 36 18 10: 10 49 31 4 40 43 22 16 7 19 37 13 28 25 46 1 11: 11 19 5 4 44 25 20 16 23 49 29 13 41 43 14 1 12: 12 42 45 30 3 36 24 33 39 9 6 21 48 15 27 18 13: 13 16 4 1 13 16 4 1 13 16 4 1 13 16 4 1 14: 14 43 41 13 29 49 23 16 20 25 44 4 5 19 11 1 15: 15 21 9 33 36 30 42 18 15 21 9 33 36 30 42 18 16: 16 1 16 1 16 1 16 1 16 1 16 1 16 1 16 1 17: 17 34 17 34 17 34 17 34 17 34 17 34 17 34 17 34 18: 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 19: 19 4 25 16 49 13 43 1 19 4 25 16 49 13 43 1 20: 20 43 44 13 5 49 11 16 14 25 41 4 29 19 23 1 21: 21 33 30 18 21 33 30 18 21 33 30 18 21 33 30 18 22: 22 25 40 13 31 19 10 16 46 43 28 4 37 49 7 1 23: 23 19 29 4 41 25 14 16 11 49 5 13 44 43 20 1 24: 24 15 3 21 45 9 12 33 27 36 48 30 6 42 39 18 25: 25 13 19 16 43 4 49 1 25 13 19 16 43 4 49 1 26: 26 13 32 16 8 4 2 1 26 13 32 16 8 4 2 1 27: 27 15 48 21 6 9 39 33 24 36 3 30 45 42 12 18 28: 28 19 22 4 10 25 37 16 40 49 46 13 7 43 31 1 29: 29 25 11 13 20 19 41 16 5 43 23 4 14 49 44 1 30: 30 33 21 18 30 33 21 18 30 33 21 18 30 33 21 18 31: 31 43 7 13 46 49 40 16 37 25 10 4 22 19 28 1 32: 32 4 26 16 2 13 8 1 32 4 26 16 2 13 8 1 33: 33 18 33 18 33 18 33 18 33 18 33 18 33 18 33 18 34: 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 35: 35 1 35 1 35 1 35 1 35 1 35 1 35 1 35 1 36: 36 21 42 33 15 30 9 18 36 21 42 33 15 30 9 18 37: 37 43 10 13 22 49 28 16 31 25 7 4 46 19 40 1 38: 38 16 47 1 38 16 47 1 38 16 47 1 38 16 47 1 39: 39 42 6 30 48 36 27 33 12 9 45 21 3 15 24 18 40: 40 19 46 4 7 25 31 16 28 49 22 13 10 43 37 1 41: 41 49 20 4 11 43 29 16 44 19 14 13 23 25 5 1 42: 42 30 36 33 9 21 15 18 42 30 36 33 9 21 15 18 43: 43 13 49 16 25 4 19 1 43 13 49 16 25 4 19 1 44: 44 49 14 4 23 43 5 16 41 19 20 13 11 25 29 1 45: 45 36 39 21 27 42 3 33 6 15 12 30 24 9 48 18 46: 46 25 28 13 37 19 7 16 22 43 40 4 31 49 10 1 47: 47 16 38 1 47 16 38 1 47 16 38 1 47 16 38 1 48: 48 9 24 30 12 15 6 33 3 42 27 21 39 36 45 18 49: 49 4 43 16 19 13 25 1 49 4 43 16 19 13 25 1 50: 50 1 50 1 50 1 50 1 50 1 50 1 50 1 50 1

52

 1 2 3 4 5 6 7 8 9 10 11 12 1: 1 1 1 1 1 1 1 1 1 1 1 1 2: 2 4 8 16 32 12 24 48 44 36 20 40 3: 3 9 27 29 35 1 3 9 27 29 35 1 4: 4 16 12 48 36 40 4 16 12 48 36 40 5: 5 25 21 1 5 25 21 1 5 25 21 1 6: 6 36 8 48 28 12 20 16 44 4 24 40 7: 7 49 31 9 11 25 19 29 47 17 15 1 8: 8 12 44 40 8 12 44 40 8 12 44 40 9: 9 29 1 9 29 1 9 29 1 9 29 1 10: 10 48 12 16 4 40 36 48 12 16 4 40 11: 11 17 31 29 7 25 15 9 47 49 19 1 12: 12 40 12 40 12 40 12 40 12 40 12 40 13: 13 13 13 13 13 13 13 13 13 13 13 13 14: 14 40 40 40 40 40 40 40 40 40 40 40 15: 15 17 47 29 19 25 11 9 31 49 7 1 16: 16 48 40 16 48 40 16 48 40 16 48 40 17: 17 29 25 9 49 1 17 29 25 9 49 1 18: 18 12 8 40 44 12 8 40 44 12 8 40 19: 19 49 47 9 15 25 7 29 31 17 11 1 20: 20 36 44 48 24 12 32 16 8 4 28 40 21: 21 25 5 1 21 25 5 1 21 25 5 1 22: 22 16 40 48 16 40 48 16 40 48 16 40 23: 23 9 51 29 43 1 23 9 51 29 43 1 24: 24 4 44 16 20 12 28 48 8 36 32 40 25: 25 1 25 1 25 1 25 1 25 1 25 1 26: 26 0 0 0 0 0 0 0 0 0 0 0 27: 27 1 27 1 27 1 27 1 27 1 27 1 28: 28 4 8 16 32 12 24 48 44 36 20 40 29: 29 9 1 29 9 1 29 9 1 29 9 1 30: 30 16 12 48 36 40 4 16 12 48 36 40 31: 31 25 47 1 31 25 47 1 31 25 47 1 32: 32 36 8 48 28 12 20 16 44 4 24 40 33: 33 49 5 9 37 25 45 29 21 17 41 1 34: 34 12 44 40 8 12 44 40 8 12 44 40 35: 35 29 27 9 3 1 35 29 27 9 3 1 36: 36 48 12 16 4 40 36 48 12 16 4 40 37: 37 17 5 29 33 25 41 9 21 49 45 1 38: 38 40 12 40 12 40 12 40 12 40 12 40 39: 39 13 39 13 39 13 39 13 39 13 39 13 40: 40 40 40 40 40 40 40 40 40 40 40 40 41: 41 17 21 29 45 25 37 9 5 49 33 1 42: 42 48 40 16 48 40 16 48 40 16 48 40 43: 43 29 51 9 23 1 43 29 51 9 23 1 44: 44 12 8 40 44 12 8 40 44 12 8 40 45: 45 49 21 9 41 25 33 29 5 17 37 1 46: 46 36 44 48 24 12 32 16 8 4 28 40 47: 47 25 31 1 47 25 31 1 47 25 31 1 48: 48 16 40 48 16 40 48 16 40 48 16 40 49: 49 9 25 29 17 1 49 9 25 29 17 1 50: 50 4 44 16 20 12 28 48 8 36 32 40 51: 51 1 51 1 51 1 51 1 51 1 51 1

55

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2: 2 4 8 16 32 9 18 36 17 34 13 26 52 49 43 31 7 14 28 1 3: 3 9 27 26 23 14 42 16 48 34 47 31 38 4 12 36 53 49 37 1 4: 4 16 9 36 34 26 49 31 14 1 4 16 9 36 34 26 49 31 14 1 5: 5 25 15 20 45 5 25 15 20 45 5 25 15 20 45 5 25 15 20 45 6: 6 36 51 31 21 16 41 26 46 1 6 36 51 31 21 16 41 26 46 1 7: 7 49 13 36 32 4 28 31 52 34 18 16 2 14 43 26 17 9 8 1 8: 8 9 17 26 43 14 2 16 18 34 52 31 28 4 32 36 13 49 7 1 9: 9 26 14 16 34 31 4 36 49 1 9 26 14 16 34 31 4 36 49 1 10: 10 45 10 45 10 45 10 45 10 45 10 45 10 45 10 45 10 45 10 45 11: 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 12: 12 34 23 1 12 34 23 1 12 34 23 1 12 34 23 1 12 34 23 1 13: 13 4 52 16 43 9 7 36 28 34 2 26 8 49 32 31 18 14 17 1 14: 14 31 49 26 34 36 9 16 4 1 14 31 49 26 34 36 9 16 4 1 15: 15 5 20 25 45 15 5 20 25 45 15 5 20 25 45 15 5 20 25 45 16: 16 36 26 31 1 16 36 26 31 1 16 36 26 31 1 16 36 26 31 1 17: 17 14 18 31 32 49 8 26 2 34 28 36 7 9 43 16 52 4 13 1 18: 18 49 2 36 43 4 17 31 8 34 7 16 13 14 32 26 28 9 52 1 19: 19 31 39 26 54 36 24 16 29 1 19 31 39 26 54 36 24 16 29 1 20: 20 15 25 5 45 20 15 25 5 45 20 15 25 5 45 20 15 25 5 45 21: 21 1 21 1 21 1 21 1 21 1 21 1 21 1 21 1 21 1 21 1 22: 22 44 33 11 22 44 33 11 22 44 33 11 22 44 33 11 22 44 33 11 23: 23 34 12 1 23 34 12 1 23 34 12 1 23 34 12 1 23 34 12 1 24: 24 26 19 16 54 31 29 36 39 1 24 26 19 16 54 31 29 36 39 1 25: 25 20 5 15 45 25 20 5 15 45 25 20 5 15 45 25 20 5 15 45 26: 26 16 31 36 1 26 16 31 36 1 26 16 31 36 1 26 16 31 36 1 27: 27 14 48 31 12 49 3 26 42 34 38 36 37 9 23 16 47 4 53 1 28: 28 14 7 31 43 49 52 26 13 34 17 36 18 9 32 16 8 4 2 1 29: 29 16 24 36 54 26 39 31 19 1 29 16 24 36 54 26 39 31 19 1 30: 30 20 50 15 10 25 35 5 40 45 30 20 50 15 10 25 35 5 40 45 31: 31 26 36 16 1 31 26 36 16 1 31 26 36 16 1 31 26 36 16 1 32: 32 34 43 1 32 34 43 1 32 34 43 1 32 34 43 1 32 34 43 1 33: 33 44 22 11 33 44 22 11 33 44 22 11 33 44 22 11 33 44 22 11 34: 34 1 34 1 34 1 34 1 34 1 34 1 34 1 34 1 34 1 34 1 35: 35 15 30 5 10 20 40 25 50 45 35 15 30 5 10 20 40 25 50 45 36: 36 31 16 26 1 36 31 16 26 1 36 31 16 26 1 36 31 16 26 1 37: 37 49 53 36 12 4 38 31 47 34 48 16 42 14 23 26 27 9 3 1 38: 38 14 37 31 23 49 47 26 53 34 27 36 48 9 12 16 3 4 42 1 39: 39 36 29 31 54 16 19 26 24 1 39 36 29 31 54 16 19 26 24 1 40: 40 5 35 25 10 15 50 20 30 45 40 5 35 25 10 15 50 20 30 45 41: 41 31 6 26 21 36 46 16 51 1 41 31 6 26 21 36 46 16 51 1 42: 42 4 3 16 12 9 48 36 27 34 53 26 47 49 23 31 37 14 38 1 43: 43 34 32 1 43 34 32 1 43 34 32 1 43 34 32 1 43 34 32 1 44: 44 11 44 11 44 11 44 11 44 11 44 11 44 11 44 11 44 11 44 11 45: 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 46: 46 26 41 16 21 31 51 36 6 1 46 26 41 16 21 31 51 36 6 1 47: 47 9 38 26 12 14 53 16 37 34 3 31 27 4 23 36 42 49 48 1 48: 48 49 42 36 23 4 27 31 3 34 37 16 53 14 12 26 38 9 47 1 49: 49 36 4 31 34 16 14 26 9 1 49 36 4 31 34 16 14 26 9 1 50: 50 25 40 20 10 5 30 15 35 45 50 25 40 20 10 5 30 15 35 45 51: 51 16 46 36 21 26 6 31 41 1 51 16 46 36 21 26 6 31 41 1 52: 52 9 28 26 32 14 13 16 7 34 8 31 17 4 43 36 2 49 18 1 53: 53 4 47 16 23 9 37 36 38 34 42 26 3 49 12 31 48 14 27 1 54: 54 1 54 1 54 1 54 1 54 1 54 1 54 1 54 1 54 1 54 1

57

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2: 2 4 8 16 32 7 14 28 56 55 53 49 41 25 50 43 29 1 3: 3 9 27 24 15 45 21 6 18 54 48 30 33 42 12 36 51 39 4: 4 16 7 28 55 49 25 43 1 4 16 7 28 55 49 25 43 1 5: 5 25 11 55 47 7 35 4 20 43 44 49 17 28 26 16 23 1 6: 6 36 45 42 24 30 9 54 39 6 36 45 42 24 30 9 54 39 7: 7 49 1 7 49 1 7 49 1 7 49 1 7 49 1 7 49 1 8: 8 7 56 49 50 1 8 7 56 49 50 1 8 7 56 49 50 1 9: 9 24 45 6 54 30 42 36 39 9 24 45 6 54 30 42 36 39 10: 10 43 31 25 22 49 34 55 37 28 52 7 13 16 46 4 40 1 11: 11 7 20 49 26 1 11 7 20 49 26 1 11 7 20 49 26 1 12: 12 30 18 45 27 39 12 30 18 45 27 39 12 30 18 45 27 39 13: 13 55 31 4 52 49 10 16 37 25 40 7 34 43 46 28 22 1 14: 14 25 8 55 29 7 41 4 56 43 32 49 2 28 50 16 53 1 15: 15 54 12 9 21 30 51 24 18 42 3 45 48 36 27 6 33 39 16: 16 28 49 43 4 7 55 25 1 16 28 49 43 4 7 55 25 1 17: 17 4 11 16 44 7 5 28 20 55 23 49 35 25 26 43 47 1 18: 18 39 18 39 18 39 18 39 18 39 18 39 18 39 18 39 18 39 19: 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 20: 20 1 20 1 20 1 20 1 20 1 20 1 20 1 20 1 20 1 21: 21 42 27 54 51 45 33 9 18 36 15 30 3 6 12 24 48 39 22: 22 28 46 43 34 7 40 25 37 16 10 49 52 4 31 55 13 1 23: 23 16 26 28 17 49 44 43 20 4 35 7 47 55 11 25 5 1 24: 24 6 30 36 9 45 54 42 39 24 6 30 36 9 45 54 42 39 25: 25 55 7 4 43 49 28 16 1 25 55 7 4 43 49 28 16 1 26: 26 49 20 7 11 1 26 49 20 7 11 1 26 49 20 7 11 1 27: 27 45 18 30 12 39 27 45 18 30 12 39 27 45 18 30 12 39 28: 28 43 7 25 16 49 4 55 1 28 43 7 25 16 49 4 55 1 29: 29 43 50 25 41 49 53 55 56 28 14 7 32 16 8 4 2 1 30: 30 45 39 30 45 39 30 45 39 30 45 39 30 45 39 30 45 39 31: 31 49 37 7 46 1 31 49 37 7 46 1 31 49 37 7 46 1 32: 32 55 50 4 14 49 29 16 56 25 2 7 53 43 8 28 41 1 33: 33 6 27 36 48 45 3 42 18 24 51 30 21 9 12 54 15 39 34: 34 16 31 28 40 49 13 43 37 4 22 7 10 55 46 25 52 1 35: 35 28 11 43 23 7 17 25 20 16 47 49 5 4 26 55 44 1 36: 36 42 30 54 6 45 24 9 39 36 42 30 54 6 45 24 9 39 37: 37 1 37 1 37 1 37 1 37 1 37 1 37 1 37 1 37 1 38: 38 19 38 19 38 19 38 19 38 19 38 19 38 19 38 19 38 19 39: 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 40: 40 4 46 16 13 7 52 28 37 55 34 49 22 25 31 43 10 1 41: 41 28 8 43 53 7 2 25 56 16 29 49 14 4 50 55 32 1 42: 42 54 45 9 36 30 6 24 39 42 54 45 9 36 30 6 24 39 43: 43 25 49 55 28 7 16 4 1 43 25 49 55 28 7 16 4 1 44: 44 55 26 4 5 49 47 16 20 25 17 7 23 43 11 28 35 1 45: 45 30 39 45 30 39 45 30 39 45 30 39 45 30 39 45 30 39 46: 46 7 37 49 31 1 46 7 37 49 31 1 46 7 37 49 31 1 47: 47 43 26 25 35 49 23 55 20 28 5 7 44 16 11 4 17 1 48: 48 24 12 6 3 30 15 36 18 9 33 45 51 54 27 42 21 39 49: 49 7 1 49 7 1 49 7 1 49 7 1 49 7 1 49 7 1 50: 50 49 56 7 8 1 50 49 56 7 8 1 50 49 56 7 8 1 51: 51 36 12 42 33 30 48 54 18 6 21 45 15 24 27 9 3 39 52: 52 25 46 55 10 7 22 4 37 43 13 49 40 28 31 16 34 1 53: 53 16 50 28 2 49 32 43 56 4 41 7 29 55 8 25 14 1 54: 54 9 30 24 42 45 36 6 39 54 9 30 24 42 45 36 6 39 55: 55 4 49 16 25 7 43 28 1 55 4 49 16 25 7 43 28 1 56: 56 1 56 1 56 1 56 1 56 1 56 1 56 1 56 1 56 1

65:

 1 2 3 4 5 6 7 8 9 10 11 12 1: 1 1 1 1 1 1 1 1 1 1 1 1 2: 2 4 8 16 32 64 63 61 57 49 33 1 3: 3 9 27 16 48 14 42 61 53 29 22 1 4: 4 16 64 61 49 1 4 16 64 61 49 1 5: 5 25 60 40 5 25 60 40 5 25 60 40 6: 6 36 21 61 41 51 46 16 31 56 11 1 7: 7 49 18 61 37 64 58 16 47 4 28 1 8: 8 64 57 1 8 64 57 1 8 64 57 1 9: 9 16 14 61 29 1 9 16 14 61 29 1 10: 10 35 25 55 30 40 10 35 25 55 30 40 11: 11 56 31 16 46 51 41 61 21 36 6 1 12: 12 14 38 1 12 14 38 1 12 14 38 1 13: 13 39 52 26 13 39 52 26 13 39 52 26 14: 14 1 14 1 14 1 14 1 14 1 14 1 15: 15 30 60 55 45 25 50 35 5 10 20 40 16: 16 61 1 16 61 1 16 61 1 16 61 1 17: 17 29 38 61 62 14 43 16 12 9 23 1 18: 18 64 47 1 18 64 47 1 18 64 47 1 19: 19 36 34 61 54 51 59 16 44 56 24 1 20: 20 10 5 35 50 25 45 55 60 30 15 40 21: 21 51 31 1 21 51 31 1 21 51 31 1 22: 22 29 53 61 42 14 48 16 27 9 3 1 23: 23 9 12 16 43 14 62 61 38 29 17 1 24: 24 56 44 16 59 51 54 61 34 36 19 1 25: 25 40 25 40 25 40 25 40 25 40 25 40 26: 26 26 26 26 26 26 26 26 26 26 26 26 27: 27 14 53 1 27 14 53 1 27 14 53 1 28: 28 4 47 16 58 64 37 61 18 49 7 1 29: 29 61 14 16 9 1 29 61 14 16 9 1 30: 30 55 25 35 10 40 30 55 25 35 10 40 31: 31 51 21 1 31 51 21 1 31 51 21 1 32: 32 49 8 61 2 64 33 16 57 4 63 1 33: 33 49 57 61 63 64 32 16 8 4 2 1 34: 34 51 44 1 34 51 44 1 34 51 44 1 35: 35 55 40 35 55 40 35 55 40 35 55 40 36: 36 61 51 16 56 1 36 61 51 16 56 1 37: 37 4 18 16 7 64 28 61 47 49 58 1 38: 38 14 12 1 38 14 12 1 38 14 12 1 39: 39 26 39 26 39 26 39 26 39 26 39 26 40: 40 40 40 40 40 40 40 40 40 40 40 40 41: 41 56 21 16 6 51 11 61 31 36 46 1 42: 42 9 53 16 22 14 3 61 27 29 48 1 43: 43 29 12 61 23 14 17 16 38 9 62 1 44: 44 51 34 1 44 51 34 1 44 51 34 1 45: 45 10 60 35 15 25 20 55 5 30 50 40 46: 46 36 31 61 11 51 6 16 21 56 41 1 47: 47 64 18 1 47 64 18 1 47 64 18 1 48: 48 29 27 61 3 14 22 16 53 9 42 1 49: 49 61 64 16 4 1 49 61 64 16 4 1 50: 50 30 5 55 20 25 15 35 60 10 45 40 51: 51 1 51 1 51 1 51 1 51 1 51 1 52: 52 39 13 26 52 39 13 26 52 39 13 26 53: 53 14 27 1 53 14 27 1 53 14 27 1 54: 54 56 34 16 19 51 24 61 44 36 59 1 55: 55 35 40 55 35 40 55 35 40 55 35 40 56: 56 16 51 61 36 1 56 16 51 61 36 1 57: 57 64 8 1 57 64 8 1 57 64 8 1 58: 58 49 47 61 28 64 7 16 18 4 37 1 59: 59 36 44 61 24 51 19 16 34 56 54 1 60: 60 25 5 40 60 25 5 40 60 25 5 40 61: 61 16 1 61 16 1 61 16 1 61 16 1 62: 62 9 38 16 17 14 23 61 12 29 43 1 63: 63 4 57 16 33 64 2 61 8 49 32 1 64: 64 1 64 1 64 1 64 1 64 1 64 1