The first few tetranacci numbers are: 0, 0, 0, 1, 1, 2, 4, 8, 15, 29, 56, 108, 208, 401, 773, 1490, 2872, 5536, 10671, 20569, 39648, 76424, 147312, 283953, 547337, … for . They represent the case of the Fibonacci n-step numbers.
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TRIBONACCI NUMBER: The tribonacci numbers are like the Fibonacci numbers, but instead of starting with two predetermined terms,
the sequence starts with three predetermined terms and each term afterwards is the sum of the preceding three terms. The first few tribonacci numbers are:
TETRANACCI NUMBER: The tetranacci numbers are a generalization of the Fibonacci numbers defined by , , , , and the recurrence relation
(1)
for . They represent the case of the Fibonacci n-step numbers . The first few terms for , 1, ... are 0, 1, 1, 2, 4, 8, 15, 29, 56, 108, 208, ….
The first few prime tetranacci numbers have indices 3, 7, 11, 12, 36, 56, 401, 2707, 8417, 14096, 31561, 50696, 53192, 155182, ... corresponding to 2, 29, 401, 773, 5350220959, with no others for
The tetranacci numbers have the generating function
(7)
The ratio of adjacent terms tends to the positive real root of , namely 1.92756...), which is sometimes known as the tetranacci constant .
The tetranacci numbers start with four predetermined terms, each term afterwards being the sum of the preceding four terms. The first few tetranacci numbers are:
0, 0, 0, 1, 1, 2, 4, 8, 15 , 29 , 56 , 108 , 208, 401, 773, 1490, 2872, 5536, 10671, 20569, 39648, 76424, 147312, 283953, 547337, …
PENTANACCI NUMBER: The pentanacci numbers are a generalization of the Fibonacci numbers defined by , , , , , and the recurrence relation
(1)
for . They represent the case of the Fibonacci n-step numbers .
The first few terms for , 2, ... are 1, 1, 2, 4, 8, 16, 31, 61, 120, 236, ... An exact formula for the th pentanacci number can be given explicitly in terms of the five roots of
(2)
as
(3)
The ratio of adjacent terms tends to the real root of , namely 1.965948236645485... sometimes called the pentanacci constant .
ACHILLES NUMBER: An Achilles number is a number that is powerful but not a perfect power. A positive integer n is a powerful number if, for every prime divisor or factor p of n, p2 is also a divisor. In other words, every prime factor appears at least squared. All Achilles numbers are powerful. However, not all powerful numbers are Achilles numbers: only those that cannot be represented as mk, where m and k are positive integers greater than 1.
Sequence of Achilles numbers
A number n = p 1 a 1 p 2 a 2 …p k a k is powerful if min(a 1 , a 2 , …, a k ) ≥ 2. If in addition gcd(a 1 , a 2 , …, a k ) = 1 the number is an Achilles number.
The smallest pair of consecutive Achilles numbers is:
5425069447 = 73 × 412 × 972
5425069448 = 23 × 260412
Examples
108 is a powerful number. Its prime factorization is 22 · 33, and thus its prime factors are 2 and 3. Both 22 = 4 and 32 = 9 are divisors of 108. However, 108 cannot be represented as mk, where m and k are positive integers greater than 1, so 108 is an Achilles number.
Finally, 784 is not an Achilles number. It is a powerful number, because not only are 2 and 7 its only prime factors, but also 22 = 4 and 72 = 49 are divisors of it. Nonetheless, it is a perfect power:
So it is not an Achilles number.
VAMPIRE NUMBERS are a type of Friedman numbers where the only operation is a multiplication of two numbers with the same number of digits, for example 1260 = 21 × 60.
Finding 2-digit Friedman numbers
There usually are fewer 2-digit Friedman numbers than 3-digit and more in any given base, but the 2-digit ones are easier to find. If we represent a 2-digit number as mb + n, where b is the base and m, n are integers from 0 to b−1, we need only check each possible combination of m and n against the equalities mb + n = mn, and mb + n = nm to see which ones are true. We need not concern ourselves with m + n or m × n, since these will always be smaller than mb + n when n < b. The same clearly holds for m − n and m/n.
DELANNOY NUMBER: In mathematics, a Delannoy number D describes the number of paths from the southwest corner (0, 0) of a rectangular grid to the northeast corner (m, n), using only single steps north, northeast, or east.
For an n × n grid, the first few Delannoy numbers (starting with n=0) are
1, 3, 13, 63, 321, 1683, 8989, 48639, 265729, ...
There are 63 Delannoy paths through a 3 × 3 grid:
The paths that do not rise above the SW–NE diagonal represent the Schröder numbers .
NARAYANA NUMBER: In combinatorics, the Narayana numbers N(n, k), n = 1, 2, 3 ..., 1 ≤ k ≤ n, form a triangular array of natural numbers, called Narayana triangle, that occur in various counting problems. They are named for T.V. Narayana (1930–1987), a mathematician from India.
The Narayana numbers can be expressed in terms of binomial coefficients :
CATALAN NUMBER: In combinatorial mathematics, the Catalan numbers form a sequence of natural numbers that occur in various counting problems, often involving recursively defined objects. The nth Catalan number is given directly in terms of binomial coefficients by
The first Catalan numbers (sequence A000108 in OEIS ) for n = 0, 1, 2, 3, … are
1 , 1, 2 , 5 , 14 , 42 , 132 , 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420, 24466267020, 91482563640, 343059613650, 1289904147324, 4861946401452,
C n is the number of different ways a convex polygon with n + 2 sides can be cut into triangles by connecting vertices with straight lines. The following hexagons illustrate the case n = 4:
FRESNEL NUMBER: The Fresnel number F, named after the physicist Augustin-Jean Fresnel, is a dimensionless number occurring in optics, in particular in diffraction theory.
For an electromagnetic wave passing through an aperture and hitting a screen, the Fresnel number F is defined as
where
is the characteristic size (e.g. radius ) of the aperture
is the distance of the screen from the aperture
is the incident wavelength .
Depending on the value of F the diffraction theory can be simplified into two special cases:
· Fraunhofer diffraction for
· Fresnel diffraction for
In case of , laws of geometrical optics are applied.
SCHRODER NUMBER: In mathematics, a Schröder number describes the number of paths from the southwest corner (0, 0) of an n × n grid to the northeast corner (n, n), using only single steps north, northeast, or east, that do not rise above the SW–NE diagonal.
The first few Schröder numbers are
1, 2, 6, 22, 90, 394, 1806, 8558,
MOTZKIN NUMBER: In mathematics, a Motzkin number for a given number n (named after Theodore Motzkin) is the number of different ways of drawing non-intersecting chords on a circle between n points. The Motzkin numbers have very diverse applications in geometry, combinatorics and number theory. The first few Motzkin numbers are:
1, 1 , 2 , 4 , 9 , 21 , 51 , 127 , 323, 835, 2188, 5798, 15511, 41835, 113634, 310572, 853467, 2356779, ……, 25669818476, 73007772802, 208023278209, 593742784829
The following figure shows the 9 ways to draw non-intersecting chords between 4 points on a circle.
The following figure shows the 21 ways to draw non-intersecting chords between 5 points on a circle.
A Motzkin prime is a Motzkin number that is prime . As of October 2007, four such primes are known:
2, 127, 15511, 953467954114363
The Motzkin number for n is also the number of positive integer sequences n−1 long in which the opening and ending elements are either 1 or 2, and the difference between any two consecutive elements is −1, 0 or 1.
Also on the upper right quadrant of a grid, the Motzkin number for n gives the number of routes from coordinate (0, 0) to coordinate (n, 0) on n steps if one is allowed to move only to the right (up, down or straight) at each step but forbidden from dipping below the y = 0 axis.
For example, the following figure shows the 9 valid Motzkin paths from (0, 0) to (4, 0):
LEYLAND NUMBER: In number theory, a Leyland number is a number of the form xy + yx, where x and y are integers greater than 1.[1] The first few Leyland numbers are
8 , 17 , 32 , 54 , 57 , 100 , 145 , 177 , 320 , 368 , 512 , 593 , 945 , 1124
The requirement that x and y both be greater than 1 is important, since without it every positive integer would be a Leyland number of the form x1 + 1x. Also, because of the commutative property of addition, the condition x ≥ y is usually added to avoid double-covering the set of Leyland numbers (so we have 1 < y ≤ x).
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As of June 2008, the largest Leyland number that has been proven to be prime is 26384405 + 44052638 with 15071 digits. From July 2004 to June 2006, it was the largest prime whose primality was proved by elliptic curve primality proving . There are many larger known probable primes such as 913829 + 991382
P(n) is the number of ways of writing n + 2 as an ordered sum in which each term is either 2 or 3 (i.e. the number of compositions of n + 2 in which each term is either 2 or 3). For example, P(6) = 4, and there are 4 ways to write 8 as an ordered sum of 2s and 3s:
2 + 2 + 2 + 2 ; 2 + 3 + 3 ; 3 + 2 + 3 ; 3 + 3 + 2
The number of ways of writing n as an ordered sum in which no term is 2 is P(2n − 2). For example, P(6) = 4, and there are 4 ways to write 4 as an ordered sum in which no term is 2:
4 ; 1 + 3 ; 3 + 1 ; 1 + 1 + 1 + 1
The number of ways of writing n as a palindromic ordered sum in which no term is 2 is P(n). For example, P(6) = 4, and there are 4 ways to write 6 as a palindromic ordered sum in which no term is 2:
6 ; 3 + 3 ; 1 + 4 + 1 ; 1 + 1 + 1 + 1 + 1 + 1
The number of ways of writing n as an ordered sum in which each term is congruent to 2 mod 3 is equal to P(n − 4). For example, P(6) = 4, and there are 4 ways to write 10 as an ordered sum in which each term is congruent to 2 mod 3:
8 + 2 ; 2 + 8 ; 5 + 5 ; 2 + 2 + 2 + 2 + 2
ICCANOBIF NUMBER: Iccanobif numbers: reverse digits of two previous terms and add
0, 1, 1, 2, 3, 5, 8, 13, 39, 124, 514, 836, 1053,
4139, 12815, 61135...
Another definition of Iccanobif numbers is: add previous two and reverse
0, 1, 1, 2, 3, 5, 8, 31, 93, 421, 415, 638, 3501,
9314, 51821, 53116...
LUCAS NUMBER: The Lucas numbers are an integer sequence named after the mathematician François Édouard Anatole Lucas (1842–1891), who studied both that sequence and the closely related Fibonacci numbers. Lucas numbers and Fibonacci numbers form complementary instances of Lucas sequences.
Like the Fibonacci numbers, each Lucas number is defined to be the sum of its two immediate previous terms, i.e. it is a Fibonacci integer sequence . Consequently, the ratio between two consecutive Lucas numbers converges to the golden ratio . However, the first two Lucas numbers are L 0 = 2 and L 1 = 1 instead of 0 and 1, and the properties of Lucas numbers are therefore somewhat different from those of Fibonacci numbers.
If L n is prime then n is either 0, prime, or a power of 2.[1] L 2 m is prime for m = 1, 2, 3, and 4 and no other known values of m .
LUCAS-CARMICHAEL NUMBER: In mathematics, a Lucas-Carmichael number is a positive composite integer n such that if p is a prime factor of n, then p + 1 is a factor of n + 1.
By convention, a number is only regarded as a Lucas-Carmichael number if it is odd and square-free (not divisible by the square of a prime number), otherwise any cube of a prime number, such as 8 or 27, would be a Lucas-Carmichael number (since n3+1 = (n+1)(n2-n+1) is always divisible by n+1).
Thus the smallest such number is 399 = 3 × 7 × 19; 399+1 = 400; 3+1, 7+1 and 19+1 are all factors of 400.
The first few numbers, and their factors, are
399 = 3 × 7 × 19
935 = 5 × 11 × 17
2015 = 5 × 13 × 31
……………
3106799 = 29 × 149 × 719
3228119 = 19 × 23 × 83 × 89
3235967 = 7 × 17 × 71 × 383
The smallest Lucas-Carmichael number with 5 factors is 588455 = 5 × 7 × 17 × 23 × 43.
PELL LUCAS NUMBER: In mathematics, the Pell numbers are an infinite sequence of integers that have been known since ancient times, the denominators of the closest rational approximations to the square root of 2. This sequence of approximations begins 1/1, 3/2, 7/5, 17/12, and 41/29, so the sequence of Pell numbers begins with 1, 2, 5, 12, and 29. The numerators of the same sequence of approximations are half the companion Pell numbers or Pell-Lucas numbers; these numbers form a second infinite sequence that begins with 2, 6, 14, 34, and 82.
GIUGA NUMBER : These are numbers n such that p divides n/p - 1 for every prime divisor p of n. Some examples are:
30, 858, 1722, 66198, 2214408306, 24423128562, 432749205173838, 14737133470010574, 550843391309130318, 244197000982499715087866346, 554079914617070801288578559178, 1910667181420507984555759916338506, ...
30 = 2*3*5 ..................... 1 = 1/2 + 1/3 + 1/5 - 1/30
858 = 2*3*11*13 ............ 1 = 1/2 + 1/3 + 1/11 + 1/13 - 1/858
1722 = 2*3*7*41 ............ 1 = 1/2 + 1/3 + 1/7 + 1/41 - 1/1722
66198 = 2*3*11*17*59 .... 1 = 1/2 + 1/3 + 1/11 + 1/17 + 1/59 - 1/66198
etc.
So far there are only 12 known Giuga numbers
[ n: p|(n/p-1) for every prime divisor p of n.]
The smallest one (= 30) has 3 prime factors (2, 3, 5)
WEIRD NUMBER: In number theory, a weird number is a natural number that is abundant but not semiperfect. In other words, the sum of the proper divisors (divisors including 1 but not itself) of the number is greater than the number, but no subset of those divisors sums to the number itself.
The smallest weird number is 70. Its proper divisors are 1, 2, 5, 7, 10, 14, and 35; these sum to 74, but no subset of these sums to 70. The number 12, for example, is abundant but not weird, because the proper divisors of 12 are 1, 2, 3, 4, and 6, which sum to 16; but 2+4+6 = 12.
The first few weird numbers are 70, 836, 4030, 5830, 7192, 7912, 9272, 10430, …It has been shown that an infinite number of weird numbers exist; in fact, the sequence of weird numbers has positive asymptotic density .
It is not known if any odd weird numbers exist; if any do, they must be greater than 232 ≈ 4×109.[4]
A large weird number is:
STROBOGRAMMATIC NUMBER: A strobogrammatic number is a number that, given a base and given a set of glyphs, appears the same whether viewed normally or upside down. In base 10, given a set of glyphs where 0, 1 and 8 are symmetrical around the horizontal axis, and 6 and 9 are the same as each other upside down, the first few strobogrammatic numbers are:
0, 1, 8, 11, 69, 88, 96, 101, 111, 181, 609, 619, 689, 808, 818, 888, 906, 916, 986, 1001, 1881 and 1961 were the most recent strobogrammatic years; the next strobogrammatic year will be 6009.
VAMPIRE NUMBER: In mathematics, a vampire number (or true vampire number) is a composite natural number v, with an even number of digits n, that can be factored into two integers x and y each with n/2 digits and not both with trailing zeroes, where v contains precisely all the digits from x and from y, in any order, counting multiplicity. x and y are called the fangs.
For example: 1260 is a vampire number, with 21 and 60 as fangs, since 21 × 60 = 1260. However, 126000 (which can be expressed as 210 × 600) is not, as both 210 and 600 have trailing zeroes. Similarly, 1023 (which can be expressed as 31 × 33) is not, because although 1023 contains all the digits of 31 and 33, the list of digits of the factors does not coincide with the list of digits of the original number.
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1260, 1395, 1435, 1530, 1827, 2187, 6880, 102510, 104260, 105210, 105264, 105750, 108135, 110758, 115672, 116725, 117067, 118440, 120600, 123354, 124483, 125248, 125433, 125460, 125500, There are many known sequences of infinitely many vampire numbers following a pattern, such as:
24959017348650 = 2947050 × 8469153 = 2949705 × 8461530 = 4125870 × 6049395 = 4129587 × 6043950 = 4230765 × 5899410
Pseudovampire numbers are similar to vampire numbers, except that the fangs of an n-digit pseudovampire number need not be of length n/2 digits. Pseudovampire numbers can have an odd number of digits, for example 126 = 6×21.
More generally, you can allow more than two fangs. In this case, vampire numbers are numbers n which can be factorized using the digits of n. For example, 1395 = 5×9×31. This sequence starts
126, 153, 688, 1206, 1255, 1260, 1395, ...
A prime vampire number, as defined by Carlos Rivera in 2002, is a true vampire number whose fangs are its prime factors. The first few prime vampire numbers are:
117067, 124483, 146137, 371893, 536539
As of 2006[update] the largest known is the square (94892254795×1045418+1)2.
ZEISEL NUMBER: A Zeisel number, named after Helmut Zeisel, is a square-free integer k with at least three prime factors which fall into the pattern
p x = ap x − 1 + b
where a and b are some integer constants and x is the index number of each prime factor in the factorization, sorted from lowest to highest. For the purpose of determining Zeisel numbers, p 0 = 1 . The first few Zeisel numbers are
105 , 1419, 1729 , 1885, 4505, 5719, 15387, 24211, 25085, 27559, 31929, 54205, 59081, 114985, 207177, 208681, 233569, 287979, 294409, 336611, 353977, 448585, 507579, 982513, 1012121, 1073305, 1242709, 1485609, 2089257, 2263811, 2953711, …
To give an example, 1729 is a Zeisel number with the constants a = 1 and b = 6, its factors being 7, 13 and 19, falling into the pattern
1729 is an example for Carmichael numbers of the kind (6n + 1)(12n + 1)(18n + 1) , which satisfies the pattern p x = ap x − 1 + b with a= 1 and b = 6n, so that every Carmichael number of the form (6n+1)(12n+1)(18n+1) is a Zeisel number.
Other Carmichael numbers of that kind are: 294409, 56052361, 118901521, 172947529, 216821881, 228842209, 1299963601, 2301745249, 9624742921, …
Hardy Ramanujan's number 1729 is also a Zeisel number.
Let be the Divisor Function of . Then two numbers and are a quasiamicable pair if
The first few are (48, 75), (140, 195), (1050, 1575), (1648, 1925), ... Quasiamicable numbers are sometimes called Betrothed Numbers or Reduced Amicable Pairs .
HONEST NUMBER: Honest numbers are numbers that can be described using exactly n letters in standard mathematical English. For example, the smallest honest numbers are 4 = "four", 8 = "two cubed", and 11 = "two plus nine". It is known that all n≥13 are honest.
Define H(n) to be the honesty number of n, the number of different ways that n can be described in exactly n letters. A number is called highly honest if H(n)=n. Are there any highly honest numbers?
Define L(n) to be the letter number of n, the minimum number of letters needed to describe n. If L(n) is less than the number of letters in the name of n, we say n is wasteful. For example, 27 is wasteful since "three cubed" is shorter than "twenty seven".
Here are a few honest numbers:
4 = four
8 = two cubed
10 = half a score; ten over one
11 = two plus nine; five plus six
13 = one plus twelve; two plus eleven; five plus eight; the sixth prime; one plus a dozen
19 = eight more than eleven; five added to fourteen; three added to sixteen; two added to seventeen; zero added to nineteen
Here are a few wasteful numbers:
24 = two dozen
27 = three cubed
48 = four dozen
100 = five score.
1 = I
2 = II
3 = III
5 = a five
6 = one six
7 = one 'n' six
9 = just a nine
12 = eleven and one
MARKOV NUMBER: A Markov number is a positive integer x, y, z that is part of a solution of the Diophantine equation:
The first few Markov numbers are:
1 , 2 , 5 , 13 , 29 , 34 , 89 , 169 , 194 , 233 , 433, 610, 985, 1325, ...
CUBAN PRIME: A cuban prime is a prime number that is a solution to one of two different specific equations involving third powers of x and y. The first of these equations is:
and the first few cuban primes from this equation are:
3, 31,331,3331,33331,333331,3333331,33333331 are prime.
But 333333331 is not prime and it is = 17 x 19607843.
Palindromic primes: Given below is a pyramid of palindromic primes:
2
30203
133020331
1713302033171
12171330203317121
151217133020331712151
1815121713302033171215181
16181512171330203317121518161
331618151217133020331712151816133
9333161815121713302033171215181613339
11933316181512171330203317121518161333911
A few 9-digit palindromic primes: An interesting list of 9-digit palindromic primes is given below:
100030001
100060001
110111011
122363221
300020003
395565593
787717787
999727999
Tetradic primes: A tetradic (or 4-way) integer is a reflectable palindromic storobogrammatic integer that is the same in four ways; i.e., whether viewed from right to left, left to right, top to bottom, or upside down. None of its digits can be other than '0,' '1,' or '8.' The first few tetradic primes are 11, 101, 181, 18181, 1008001, and 1180811
Arithmetic Progression: The standard arithmetic progression, which is well known is:
A n = a + (a+d) + (a+2d)+……….{a+(n-1)d}= [n(2a + (n-1)d]/2
Arithmeticprogression Type I:
A n = a + (a+d+r) + (a+2d+ r2) +…..(a+(n-1)d+ rn-1), where the successive terms differ also by the terms of a G.P.
One can see that this is a combination of A.P + G.P.
A n = [2a + (n-1)d]n/2 + a.[rn -1]/(r-1)
As n tends to infinity, A n also tends to infinity, in both cases, whether mod r is <1 or > 1, since n occurs in the A.P. sum.
Geometric progression: The standard geometric progression which is well known is:
G n = a + ar + ar2 + ar3 +………..+ arn-1 = [a(rn – 1]/[r-1] if r > 1.
= [a(1-rn)/(1-r)] if mod r <1.
If r >1, as n tends to infinity, G n tends also to infinity.
As n tends to infinity if mod r < 1, G n tends to [a/(r-1)]
Geometric progression type I: In this, the successive terms differ by a number which varies in AP from term to term:
G 1n = a + (ar + d) + (ar2 + 2d) + (ar3 + 3d)+….(arn-1 + (n-1)d)
= a + ar + ar2 + ar3 +…….+arn-1 + d +2d + 3d +……..+(n-1)d
= [a(1-rn)/(1-r)] + d (1 + 2 + …..+ (n-1)
= [a(1-rn)/(1-r)] + [d(n-1)n/2]
If d > 0, we may call such series as abundant geometric series, and if
d < 0, we may call such series as deficient or defective geometric series.
Harmonic progression:
H n = (1/a) + {1/(a+d)} + {1/(a+2d)}+ {1/(a+3d)}+….. {1/(a+(n-1)d)}
There is no easy formula for the summation of this progression.
Arithmetico Geometric Series: AG n = a + (a+d)r + (a+2d)r2 +…………
Awaken your dormant DNA ability to attract wealth effortlessly
The simple yet scientifically proven Wealth DNA method laid out in the report allows you to effortlessly start attracting the wealth and abundance you deserve.
Awaken your dormant DNA ability to attract wealth effortlessly
The simple yet scientifically proven Wealth DNA method laid out in the report allows you to effortlessly start attracting the wealth and abundance you deserve.
Affluent Savvy
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