| Type: | Package |
| Title: | Integer Sequence Generator |
| Version: | 0.2.1 |
| Description: | Generates well-known integer sequences. 'gmp' package is adopted for computing with arbitrarily large numbers. Every function has hyperlink to its corresponding item in OEIS (The On-Line Encyclopedia of Integer Sequences) in the function help page. For interested readers, see Sloane and Plouffe (1995, ISBN:978-0125586306). |
| License: | MIT + file LICENSE |
| Imports: | gmp, utils |
| Encoding: | UTF-8 |
| RoxygenNote: | 7.2.1 |
| NeedsCompilation: | no |
| Packaged: | 2022-09-06 21:18:30 UTC; kisung |
| Author: | Kisung You  [aut,
cre] |
| Maintainer: | Kisung You <kisungyou@outlook.com> |
| Repository: | CRAN |
| Date/Publication: | 2022-09-07 07:50:18 UTC |
Zseq : Integer Sequence Generator
Description
The world of integer sequence has long history, which has been accumulated in The On-Line Encyclopedia of Integer Sequences.
Even though R is not a first pick for many number theorists, we introduce our package to enrich the R ecosystem as well as
provide pedagogical toolset. We adopted gmp for flexible large number computations in that users can easily experience
large number sequences on a non-exclusive generic computing platform.
Abundant numbers
Description
Under OEIS A005101, an abundant number is a number whose proper divisors sum up to the extent greater than the number itself. First 6 abundant numbers are 12, 18, 20, 24, 30, 36.
Usage
Abundant(n, gmp = TRUE)
Arguments
n |
the number of first |
gmp |
a logical; |
Value
a vector of length n containing first entries from the sequence.
See Also
Examples
## generate first 30 Abundant numbers and print it
print(Abundant(30))
Achilles numbers
Description
Under OEIS A052486, an Achilles number is a number that is powerful but not perfect. First 6 Achilles numbers are 72, 108, 200, 288, 392, 432.
Usage
Achilles(n, gmp = TRUE)
Arguments
n |
the number of first |
gmp |
a logical; |
Value
a vector of length n containing first entries from the sequence.
Examples
## generate first 3 Achilles numbers and print
print(Achilles(3))
Bell numbers
Description
Under OEIS A000110, the nth Bell number is
the number of ways to partition a set of n labeled elements,
where the first 6 entries are 1, 1, 2, 5, 15, 52.
Usage
Bell(n, gmp = TRUE)
Arguments
n |
the number of first |
gmp |
a logical; |
Value
a vector of length n containing first entries from the sequence.
Examples
## generate first 30 Bell numbers and print
print(Bell(30))
Carmichael numbers
Description
Under OEIS A002997, a Carmichael number is
a composite number n such that
b^{n-1} = 1 (mod n)
for all integers b which are relatively prime to n. First 6 Carmichael numbers are 561, 1105, 1729, 2465, 2821, 6601.
Usage
Carmichael(n, gmp = TRUE)
Arguments
n |
the number of first |
gmp |
a logical; |
Value
a vector of length n containing first entries from the sequence.
Examples
## generate first 3 Carmichael numbers
print(Carmichael(3))
Catalan numbers
Description
Under OEIS A000108, the nth Catalan number is given as
C_n = \frac{(2n)!}{(n+1)!n!}
where the first 6 entries are 1, 1, 2, 5, 14, 42 with n\ge 0.
Usage
Catalan(n, gmp = TRUE)
Arguments
n |
the number of first |
gmp |
a logical; |
Value
a vector of length n containing first entries from the sequence.
Examples
## generate first 30 Catalan numbers
print(Catalan(30))
Composite numbers
Description
Under OEIS A002808, a composite number is a positive integer that can be represented as multiplication of two smaller positive integers. The first 6 composite numbers are 4, 6, 8, 9, 10, 12.
Usage
Composite(n, gmp = TRUE)
Arguments
n |
the number of first |
gmp |
a logical; |
Value
a vector of length n containing first entries from the sequence.
Examples
## generate first 30 Composite numbers
print(Composite(30))
Deficient numbers
Description
Under OEIS A005100, a deficient number is a number whose proper divisors sum up to the extent smaller than the number itself. First 6 deficient numbers are 1, 2, 3, 4, 5, 7
Usage
Deficient(n, gmp = TRUE)
Arguments
n |
the number of first |
gmp |
a logical; |
Value
a vector of length n containing first entries from the sequence.
See Also
Examples
## generate first 30 Deficient numbers
print(Deficient(30))
Equidigital numbers
Description
Under OEIS A046758, an Equidigital number has equal digits as the number of digits in its prime factorization including exponents. First 6 Equidigital numbers are 1, 2, 3, 5, 7, 10. Though it doesn't matter which base we use, here we adopt only a base of 10.
Usage
Equidigital(n, gmp = TRUE)
Arguments
n |
the number of first |
gmp |
a logical; |
Value
a vector of length n containing first entries from the sequence.
See Also
Examples
## generate first 20 Equidigital numbers
print(Equidigital(20))
Evil numbers
Description
Under OEIS A001969, an Evil number has an even number of 1's in its binary expansion. First 6 Evil numbers are 0, 3, 5, 6, 9, 10.
Usage
Evil(n, gmp = TRUE)
Arguments
n |
the number of first |
gmp |
a logical; |
Value
a vector of length n containing first entries from the sequence.
See Also
Examples
## generate first 20 Evil numbers
print(Evil(20))
Extravagant numbers
Description
Under OEIS A046760, an Extravagant number has less digits than the number of digits in its prime factorization including exponents. First 6 Extravagant numbers are 4, 6, 8, 9, 12, 18. Though it doesn't matter which base we use, here we adopt only a base of 10.
Usage
Extravagant(n, gmp = TRUE)
Arguments
n |
the number of first |
gmp |
a logical; |
Value
a vector of length n containing first entries from the sequence.
See Also
Examples
## generate first 20 Extravagant numbers
print(Extravagant(20))
Factorial numbers
Description
Under OEIS A000142, a Factorial is the product of all positive integers smaller than or equal to the number. First 6 such numbers are 1, 1, 2, 6, 24, 120
Usage
Factorial(n, gmp = TRUE)
Arguments
n |
the number of first |
gmp |
a logical; |
Value
a vector of length n containing first entries from the sequence.
Examples
## generate first 10 Factorials
print(Factorial(10))
Alternating Factorial numbers
Description
Under OEIS A005165, an Alternating Factorial is the absolute value of the alternating sum of the
first n factorials of positive integers. First 6 such numbers are 0, 1, 1, 5, 19, 101.
Usage
Factorial.Alternating(n, gmp = TRUE)
Arguments
n |
the number of first |
gmp |
a logical; |
Value
a vector of length n containing first entries from the sequence.
See Also
Examples
## generate first 5 Alternating Factorial numbers
print(Factorial.Alternating(5))
Double Factorial numbers
Description
Under OEIS A000165 and A001147, a Double Factorial is the factorial of numbers with same parity.
For example, if n=5, then n!!=5*3*1.
Usage
Factorial.Double(n, gmp = TRUE, odd = TRUE)
Arguments
n |
the number of first |
gmp |
a logical; |
odd |
a logical; |
Value
a vector of length n containing first entries from the sequence.
See Also
Examples
## generate first 10 double factorials
print(Factorial.Double(10))
Fibonacci numbers
Description
Under OEIS A000045, the nth Fibonnaci number is given as
F_n = F_{n-1} + F_{n-2}
where the first 6 entries are 0, 1, 1, 2, 3, 5 with n\ge 0.
Usage
Fibonacci(n, gmp = TRUE)
Arguments
n |
the number of first |
gmp |
a logical; |
Value
a vector of length n containing first entries from the sequence.
Examples
## generate first 30 Fibonacci numbers
print(Fibonacci(30))
Frugal numbers
Description
Under OEIS A046759, a Frugal number has more digits than the number of digits in its prime factorization including exponents. First 6 Frugal numbers are 125, 128, 243, 256, 343, 512. Though it doesn't matter which base we use, here we adopt only a base of 10.
Usage
Frugal(n, gmp = TRUE)
Arguments
n |
the number of first |
gmp |
a logical; |
Value
a vector of length n containing first entries from the sequence.
See Also
Examples
## generate first 5 Frugal numbers
print(Frugal(5))
Happy numbers
Description
Under OEIS A007770, a Happy number is defined by the process that starts from arbitrary positive integer and replaces the number by the sum of the squares of each digit until the number is 1. First 6 Happy numbers are 1, 7, 10, 13, 19, 23.
Usage
Happy(n, gmp = TRUE)
Arguments
n |
the number of first |
gmp |
a logical; |
Value
a vector of length n containing first entries from the sequence.
Examples
## generate first 30 happy numbers
print(Happy(30))
Juggler sequence
Description
Under OEIS A094683, a Juggler sequence is an integer-valued
sequence that starts with a nonnegative number iteratively follows that J_{k+1}=floor(J_k^{1/2}) if J_k is even, or J_{k+1}=floor(J_k^{3/2}) if odd.
No first 6 terms are given since it all depends on the starting value.
Usage
Juggler(start, gmp = TRUE)
Arguments
start |
the starting nonnegative integer. |
gmp |
a logical; |
Value
a vector recording the sequence of unknown length a priori.
Examples
## let's start from 9 and show the sequence
print(Juggler(9))
Largest value for Juggler sequence
Description
Under OEIS A094716, the Largest value for Juggler sequence is the largest value in trajectory of
a sequence that starts from n. First 6 terms are 0, 1, 2, 36, 4, 36 that n starting from 0 is conventional choice.
Usage
Juggler.Largest(n, gmp = TRUE)
Arguments
n |
the number of first |
gmp |
a logical; |
Value
a vector of length n containing first entries from the sequence.
See Also
Examples
## generate first 10 numbers of largest values for Juggler sequences
print(Juggler.Largest(10))
Number of steps for Juggler sequence
Description
Under OEIS A007320, a Number of steps for Juggler sequence literally counts the number of steps
required for a sequence that starts from n. First 6 terms are 0, 1, 6, 2, 5, 2 that n starting from 0 is conventional choice.
Note that when it counts number of steps, not the length of the sequence including the last 1.
Usage
Juggler.Nsteps(n, gmp = TRUE)
Arguments
n |
the number of first |
gmp |
a logical; |
Value
a vector of length n containing first entries from the sequence.
See Also
Examples
## generate first 10 numbers of steps for Juggler sequences
print(Juggler.Nsteps(10))
Lucas numbers
Description
Under OEIS A000032, the nth Lucas number is given as
F_n = F_{n-1} + F_{n-2}
where the first 6 entries are 2, 1, 3, 4, 7, 11.
Usage
Lucas(n, gmp = TRUE)
Arguments
n |
the number of first |
gmp |
a logical; |
Value
a vector of length n containing first entries from the sequence.
See Also
Examples
## generate first 30 Lucas numbers
print(Lucas(30))
Motzkin numbers
Description
Under OEIS A001006, a Motzkin number for a given n is the number of ways for drawing non-intersecting chords among n points on a circle, where the first 7 entries are 1, 1, 2, 4, 9, 21, 51.
Usage
Motzkin(n, gmp = TRUE)
Arguments
n |
the number of first |
gmp |
a logical; |
Value
a vector of length n containing first entries from the sequence.
Examples
## generate first 30 Motzkin numbers
print(Motzkin(30))
Odious numbers
Description
Under OEIS A000069, an Odious number has an odd number of 1's in its binary expansion. First 6 Odious numbers are 1, 2, 4, 7, 8, 11.
Usage
Odious(n, gmp = TRUE)
Arguments
n |
the number of first |
gmp |
a logical; |
Value
a vector of length n containing first entries from the sequence.
See Also
Examples
## generate first 20 Odious numbers
print(Odious(20))
Padovan numbers
Description
Under OEIS A000931, the nth Padovan number is given as
F_n = F_{n-2} + F_{n-3}
where the first 6 entries are 1, 0, 0, 1, 0, 1.
Usage
Padovan(n, gmp = TRUE)
Arguments
n |
the number of first |
gmp |
a logical; |
Value
a vector of length n containing first entries from the sequence.
Examples
## generate first 30 Padovan numbers
print(Padovan(30))
Palindromic numbers
Description
Under OEIS A002113, a Palindromic number is a number that
remains the same when its digits are reversed. First 6 Palindromic numbers in decimal are 0, 1, 2, 3, 4, 5.
This function supports various base by specifying the parameter base but returns are still represented in decimal.
Usage
Palindromic(n, base = 10, gmp = TRUE)
Arguments
n |
the number of first |
base |
choice of base. |
gmp |
a logical; |
Value
a vector of length n containing first entries from the sequence.
Examples
## generate first 30 palindromic number in decimal
print(Palindromic(30))
Palindromic squares
Description
Under OEIS A002779, a Palindromic square is a number that is both Palindromic and Square. First 6 such numbers are 0, 1, 4, 9, 121, 484. It uses only the base 10 decimals.
Usage
Palindromic.Squares(n, gmp = TRUE)
Arguments
n |
the number of first |
gmp |
a logical; |
Value
a vector of length n containing first entries from the sequence.
Examples
## generate first 10 palindromic squares
print(Palindromic.Squares(10))
Perfect numbers
Description
Under OEIS A000396, a Perfect number is a number whose proper divisors sum up to the extent equal to the number itself. First 6 abundant numbers are 6, 28, 496, 8128, 33550336, 8589869056.
Usage
Perfect(n, gmp = TRUE)
Arguments
n |
the number of first |
gmp |
a logical; |
Value
a vector of length n containing first entries from the sequence.
See Also
Examples
## Not run:
## generate first 7 Perfect numbers
print(Perfect(10))
## End(Not run)
Perrin numbers
Description
Under OEIS A001608, the nth Perrin number is given as
F_n = F_{n-2} + F_{n-3}
where the first 6 entries are 3, 0, 2, 3, 2, 5.
Usage
Perrin(n, gmp = TRUE)
Arguments
n |
the number of first |
gmp |
a logical; |
Value
a vector of length n containing first entries from the sequence.
Examples
## generate first 30 Perrin numbers
print(Perrin(30))
Powerful numbers
Description
Under OEIS A001694, a Powerful number is a positive integer such that
for every prime p dividing the number, p^2 also divides the number. First
6 powerful numbers are 1, 4, 8, 9, 16, 25.
Usage
Powerful(n, gmp = TRUE)
Arguments
n |
the number of first |
gmp |
a logical; |
Value
a vector of length n containing first entries from the sequence.
Examples
## generate first 20 Powerful numbers
print(Powerful(20))
Prime numbers
Description
Under OEIS A000040, a Prime number is a natural number with no positive divisors other than 1 and itself. First 6 prime numbers are 2, 3, 5, 7, 11, 13.
Usage
Prime(n, gmp = TRUE)
Arguments
n |
the number of first |
gmp |
a logical; |
Value
a vector of length n containing first entries from the sequence.
Examples
## generate first 30 Regular numbers
print(Prime(30))
Regular numbers
Description
Under OEIS A051037, a Regular number - also known as 5-smooth - is a positive integer that even divide powers of 60, or equivalently, whose prime divisors are only 2,3, and 5. First 6 Regular numbers are 1, 2, 3, 4, 5, 6.
Usage
Regular(n, gmp = TRUE)
Arguments
n |
the number of first |
gmp |
a logical; |
Value
a vector of length n containing first entries from the sequence.
Examples
## generate first 20 Regular numbers
print(Regular(20))
Square numbers
Description
Under OEIS A000290, a Square number is
A_n = n^2
for n\ge 0. First 6 Square numbers are 0, 1, 4, 9, 16, 25.
Usage
Square(n, gmp = TRUE)
Arguments
n |
the number of first |
gmp |
a logical; |
Value
a vector of length n containing first entries from the sequence.
Examples
## generate first 20 Square numbers
print(Square(20))
Squarefree numbers
Description
Under OEIS A005117, a Squarefree number is a number that are not divisible by a square of a smaller integer greater than 1. First 6 Squarefree numbers are 1, 2, 3, 5, 6, 7.
Usage
Squarefree(n, gmp = TRUE)
Arguments
n |
the number of first |
gmp |
a logical; |
Value
a vector of length n containing first entries from the sequence.
Examples
## generate first 30 Squarefree numbers
print(Squarefree(30))
Telephone numbers
Description
Under OEIS A000085, a Telephone number - also known as Involution number - is counting the number
of connection patterns in a telephone system with n subscribers, or in a more mathematical term, the number of self-inverse permutations on
n letters. First 6 Telephone numbers are 1, 1, 2, 4, 10, 26,
Usage
Telephone(n, gmp = TRUE)
Arguments
n |
the number of first |
gmp |
a logical; |
Value
a vector of length n containing first entries from the sequence.
Examples
## generate first 20 Regular numbers
print(Telephone(20))
Thabit numbers
Description
Under OEIS A055010, the nth Thabit number is given as
A_n = 3*2^{n-1}-1
where the first 6 entries are 0, 2, 5, 11, 23, 47 with A_0 = 0.
Usage
Thabit(n, gmp = TRUE)
Arguments
n |
the number of first |
gmp |
a logical; |
Value
a vector of length n containing first entries from the sequence.
Examples
## generate first 30 Thabit numbers
print(Thabit(30))
Triangular numbers
Description
Under OEIS A000217, a Triangular number counts objects arranged in an equilateral triangle. First 6 Triangular numbers are 0, 1, 3, 6, 10, 15.
Usage
Triangular(n, gmp = TRUE)
Arguments
n |
the number of first |
gmp |
a logical; |
Value
a vector of length n containing first entries from the sequence.
Examples
## generate first 20 Triangular numbers
print(Triangular(20))
Unusual numbers
Description
Under OEIS A064052, an Unusual number is a natural number whose largest prime factor is strictly greater than square root of the number. First 6 Unusual numbers are 2, 3, 5, 6, 7, 10.
Usage
Unusual(n, gmp = TRUE)
Arguments
n |
the number of first |
gmp |
a logical; |
Value
a vector of length n containing first entries from the sequence.
Examples
## generate first 20 Unusual numbers
print(Unusual(20))