Package 'Zseq'

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.

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

Value

a vector of length n containing first entries from the sequence.

See Also

Deficient, Perfect

Examples

## generate first 30 Abundant numbers and print it
print(Abundant(30))

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

Value

a vector of length n containing first entries from the sequence.

Examples

## generate first 3 Achilles numbers and print
print(Achilles(3))

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

Value

a vector of length n containing first entries from the sequence.

Examples

## generate first 30 Bell numbers and print
print(Bell(30))

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

Value

a vector of length n containing first entries from the sequence.

Examples

## generate first 3 Carmichael numbers
print(Carmichael(3))

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

Value

a vector of length n containing first entries from the sequence.

Examples

## generate first 30 Catalan numbers
print(Catalan(30))

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

Value

a vector of length n containing first entries from the sequence.

Examples

## generate first 30 Composite numbers
print(Composite(30))

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

Value

a vector of length n containing first entries from the sequence.

See Also

Abundant, Perfect

Examples

## generate first 30 Deficient numbers
print(Deficient(30))

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

Value

a vector of length n containing first entries from the sequence.

See Also

Frugal, Extravagant

Examples

## generate first 20 Equidigital numbers
print(Equidigital(20))

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

Value

a vector of length n containing first entries from the sequence.

See Also

Odious

Examples

## generate first 20 Evil numbers
print(Evil(20))

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

Value

a vector of length n containing first entries from the sequence.

See Also

Frugal, Equidigital

Examples

## generate first 20 Extravagant numbers
print(Extravagant(20))

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

Value

a vector of length n containing first entries from the sequence.

Examples

## generate first 10 Factorials
print(Factorial(10))

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

Value

a vector of length n containing first entries from the sequence.

See Also

Factorial

Examples

## generate first 5 Alternating Factorial numbers
print(Factorial.Alternating(5))

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

Value

a vector of length n containing first entries from the sequence.

See Also

Factorial

Examples

## generate first 10 double factorials
print(Factorial.Double(10))

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

Value

a vector of length n containing first entries from the sequence.

Examples

## generate first 30 Fibonacci numbers
print(Fibonacci(30))

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

Value

a vector of length n containing first entries from the sequence.

See Also

Extravagant, Equidigital

Examples

## generate first 5 Frugal numbers
print(Frugal(5))

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

Value

a vector of length n containing first entries from the sequence.

Examples

## generate first 30 happy numbers
print(Happy(30))

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

Value

a vector recording the sequence of unknown length a priori.

Examples

## let's start from 9 and show the sequence
print(Juggler(9))

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

Value

a vector of length n containing first entries from the sequence.

See Also

Juggler

Examples

## generate first 10 numbers of largest values for Juggler sequences
print(Juggler.Largest(10))

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

Value

a vector of length n containing first entries from the sequence.

See Also

Juggler

Examples

## generate first 10 numbers of steps for Juggler sequences
print(Juggler.Nsteps(10))

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

Value

a vector of length n containing first entries from the sequence.

See Also

Fibonacci

Examples

## generate first 30 Lucas numbers
print(Lucas(30))

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

Value

a vector of length n containing first entries from the sequence.

Examples

## generate first 30 Motzkin numbers
print(Motzkin(30))

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

Value

a vector of length n containing first entries from the sequence.

See Also

Evil

Examples

## generate first 20 Odious numbers
print(Odious(20))

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

Value

a vector of length n containing first entries from the sequence.

Examples

## generate first 30 Padovan numbers
print(Padovan(30))

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

Value

a vector of length n containing first entries from the sequence.

Examples

## generate first 30 palindromic number in decimal
print(Palindromic(30))

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

Value

a vector of length n containing first entries from the sequence.

Examples

## generate first 10 palindromic squares
print(Palindromic.Squares(10))

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

Value

a vector of length n containing first entries from the sequence.

See Also

Deficient, Abundant

Examples

## Not run: 
## generate first 7 Perfect numbers
print(Perfect(10))

## End(Not run)

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

Value

a vector of length n containing first entries from the sequence.

Examples

## generate first 30 Perrin numbers
print(Perrin(30))

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

Value

a vector of length n containing first entries from the sequence.

Examples

## generate first 20 Powerful numbers
print(Powerful(20))

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

Value

a vector of length n containing first entries from the sequence.

Examples

## generate first 30 Regular numbers
print(Prime(30))

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

Value

a vector of length n containing first entries from the sequence.

Examples

## generate first 20 Regular numbers
print(Regular(20))

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

Value

a vector of length n containing first entries from the sequence.

Examples

## generate first 20 Square numbers
print(Square(20))

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

Value

a vector of length n containing first entries from the sequence.

Examples

## generate first 30 Squarefree numbers
print(Squarefree(30))

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

Value

a vector of length n containing first entries from the sequence.

Examples

## generate first 20 Regular numbers
print(Telephone(20))

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

Value

a vector of length n containing first entries from the sequence.

Examples

## generate first 30 Thabit numbers
print(Thabit(30))

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

Value

a vector of length n containing first entries from the sequence.

Examples

## generate first 20 Triangular numbers
print(Triangular(20))

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

Value

a vector of length n containing first entries from the sequence.

Examples

## generate first 20 Unusual numbers
print(Unusual(20))