In number theory, a perfect digital invariant (PDI) is a number in a given number base ( b {\displaystyle b} ) that is the sum of its own digits each...

{\displaystyle b} that eventually reaches 1 when iterated over the perfect digital invariant function for p = 2 {\displaystyle p=2} . The origin of happy numbers...

(also known as a pluperfect digital invariant (PPDI), an Armstrong number (after Michael F. Armstrong) or a plus perfect number) in a given number base...

Arithmetic dynamics Casting out nines Checksum Digital root Hamming weight Harshad number Perfect digital invariant Sideways sum Smith number Sum-product number...

Kaprekar number Meertens number Narcissistic number Perfect digit-to-digit invariant Perfect digital invariant Sum-product number Sorting algorithm Hanover 2017...

In mathematics, a square number or perfect square is an integer that is the square of an integer; in other words, it is the product of some integer with...

because all perfect cubes must have digital root 1, 8 or 9. That is their values modulo 9 may be only 0, 1, and 8. Moreover, the digital root of any number's...

mathematics, a multiply perfect number (also called multiperfect number or pluperfect number) is a generalization of a perfect number. For a given natural...

In number theory, a perfect number is a positive integer that is equal to the sum of its positive proper divisors, that is, divisors excluding the number...

be a perfect number. More generally, no Fibonacci number other than 1 can be multiply perfect, and no ratio of two Fibonacci numbers can be perfect. With...

Kaprekar number Meertens number Narcissistic number Perfect digit-to-digit invariant Perfect digital invariant Sloane, N. J. A. (ed.). "Sequence A038369 (Numbers...

The digital root (also repeated digital sum) of a natural number in a given radix is the (single digit) value obtained by an iterative process of summing...

In number theory, a perfect digit-to-digit invariant (PDDI; also known as a Munchausen number) is a natural number in a given number base b {\displaystyle...

and the fundamental theorem of arithmetic, and shows how to construct a perfect number from a Mersenne prime. Another Greek invention, the Sieve of Eratosthenes...

In mathematics, an almost perfect number (sometimes also called slightly defective or least deficient number) is a natural number n such that the sum...

prime factors of a number are repeated it is called a powerful number (All perfect powers are powerful numbers). If none of its prime factors are repeated...

where Mp is a Mersenne prime. No odd perfect numbers are known; hence, all known perfect numbers are triangular. For example, the third triangular...

units from indefinitely many units is a 2). However, in the definition of perfect number which comes shortly afterward, Euclid treats 1 as a number like...

In mathematics, a perfect power is a natural number that is a product of equal natural factors, or, in other words, an integer that can be expressed as...

Meertens Other Dudeney Factorion Kaprekar Kaprekar's constant Keith Lychrel Narcissistic Perfect digit-to-digit invariant Perfect digital invariant Happy...

numbers are perfect numbers with σ(n) = 2n, and abundant numbers with σ(n) > 2n. Nicomachus was the first to subdivide numbers into deficient, perfect, or abundant...

k)^{2+\epsilon }} for sufficiently large k. Every multiple of a perfect number (except the perfect number itself) is abundant. For example, every multiple of...

recurring digital representations of unit fractions. A cyclic number of length L is the digital representation of 1/(L + 1). Conversely, if the digital period...

Meertens Other Dudeney Factorion Kaprekar Kaprekar's constant Keith Lychrel Narcissistic Perfect digit-to-digit invariant Perfect digital invariant Happy...

Meertens Other Dudeney Factorion Kaprekar Kaprekar's constant Keith Lychrel Narcissistic Perfect digit-to-digit invariant Perfect digital invariant Happy...

constant Kaprekar number Narcissistic number Perfect digit-to-digit invariant Perfect digital invariant Sum-product number Richard S. Bird (1998). "Meertens...

have some other property are listed below: There are many palindromic perfect powers nk, where n is a natural number and k is 2, 3 or 4. Palindromic...

persistence Digital root Digit product dynamics Multiplicative digital root Multiplicative persistence Lychrel number Perfect digital invariant Happy number...

Meertens Other Dudeney Factorion Kaprekar Kaprekar's constant Keith Lychrel Narcissistic Perfect digit-to-digit invariant Perfect digital invariant Happy...

Exponentiation Fifth power (algebra) Sixth power Seventh power Eighth power Perfect power An odd fourth power is the square of an odd square number. All odd...