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- Primary_pseudoperfect_number abstract "In mathematics, and particularly in number theory, a primary pseudoperfect number is a number N that satisfies the Egyptian fraction equationwhere the sum is over only the prime divisors of N. Equivalently (as can be seen by multiplying this equation by N),Except for the exceptional primary pseudoperfect number 2, this expression gives a representation for N as a sum of a set of distinct divisors of N; therefore each such number (except 2) is pseudoperfect.Primary pseudoperfect numbers were first investigated and named by Butske, Jaje, and Mayernik (2000). The first few primary pseudoperfect numbers are2, 6, 42, 1806, 47058, 2214502422, 52495396602, ... (sequence A054377 in OEIS).The first four of these numbers are one less than the corresponding numbers in Sylvester's sequence but later numbers in Sylvester's sequence do not similarly correspond to primary pseudoperfect numbers. It is unknown whether there are infinitely many primary pseudoperfect numbers, or whether there are any odd primary pseudoperfect numbers.The prime factors of primary pseudoperfect numbers may provide solutions to Znám's problem in which all members of the solution set are prime. For instance, the factors of the primary pseudoperfect number 47058 are the solution set {2,3,11,23,31} to Znám's problem. However, the smaller primary pseudoperfect numbers 2, 6, 42, and 1806 do not correspond to solutions to Znám's problem in this way, as their sets of prime factors violate the requirement in Znám's problem that no number in the set can equal one plus the product of all the other numbers. Anne (1998) observes that there is exactly one solution set of this type that has k primes in it, for each k ≤ 8, and conjectures that the same is true for larger k.If a primary pseudoperfect number N is one less than a prime number, then N×(N+1) is also primary pseudoperfect. For instance, 47058 is primary pseudoperfect, and 47059 is prime, so 47058 × 47059 = 2214502422 is also primary pseudoperfect.See also Giuga number.".
- Primary_pseudoperfect_number thumbnail Znam-2-3-11-23-31.svg?width=300.
- Primary_pseudoperfect_number wikiPageID "7184831".
- Primary_pseudoperfect_number wikiPageRevisionID "557557049".
- Primary_pseudoperfect_number hasPhotoCollection Primary_pseudoperfect_number.
- Primary_pseudoperfect_number title "Primary Pseudoperfect Number".
- Primary_pseudoperfect_number urlname "PrimaryPseudoperfectNumber".
- Primary_pseudoperfect_number subject Category:Egyptian_fractions.
- Primary_pseudoperfect_number subject Category:Integer_sequences.
- Primary_pseudoperfect_number type Abstraction100002137.
- Primary_pseudoperfect_number type Arrangement107938773.
- Primary_pseudoperfect_number type Chemical114806838.
- Primary_pseudoperfect_number type EgyptianFractions.
- Primary_pseudoperfect_number type Fraction114922107.
- Primary_pseudoperfect_number type Group100031264.
- Primary_pseudoperfect_number type IntegerSequences.
- Primary_pseudoperfect_number type Material114580897.
- Primary_pseudoperfect_number type Matter100020827.
- Primary_pseudoperfect_number type Ordering108456993.
- Primary_pseudoperfect_number type Part113809207.
- Primary_pseudoperfect_number type PhysicalEntity100001930.
- Primary_pseudoperfect_number type Relation100031921.
- Primary_pseudoperfect_number type Sequence108459252.
- Primary_pseudoperfect_number type Series108457976.
- Primary_pseudoperfect_number type Substance100019613.
- Primary_pseudoperfect_number comment "In mathematics, and particularly in number theory, a primary pseudoperfect number is a number N that satisfies the Egyptian fraction equationwhere the sum is over only the prime divisors of N.".
- Primary_pseudoperfect_number label "Primary pseudoperfect number".
- Primary_pseudoperfect_number sameAs m.025vgrg.
- Primary_pseudoperfect_number sameAs Q7243159.
- Primary_pseudoperfect_number sameAs Q7243159.
- Primary_pseudoperfect_number sameAs Primary_pseudoperfect_number.
- Primary_pseudoperfect_number wasDerivedFrom Primary_pseudoperfect_number?oldid=557557049.
- Primary_pseudoperfect_number depiction Znam-2-3-11-23-31.svg.
- Primary_pseudoperfect_number isPrimaryTopicOf Primary_pseudoperfect_number.
