Erdős–Nicolas number

Erdős–Nicolas number
Named after	Paul Erdős, Jean-Louis Nicolas
Publication year	1975
Author of publication	Erdős, P., Nicolas, J. L.
Subsequence of	Abundant numbers
First terms	24, 2016, 8190
Largest known term	3304572752464376776401640967110656
OEIS index	A194472; Erdős-Nicolas numbers;

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In number theory, an Erdős–Nicolas number is a number that is not perfect, but that equals one of the partial sums of its divisors. That is, a number $n$ is an Erdős–Nicolas number when there exists another number $m$ such that

\sum _{d\mid n,\ d\leq m}d=n.

^[1]

The first ten Erdős–Nicolas numbers are

24, 2016, 8190, 42336, 45864, 392448, 714240, 1571328, 61900800 and 91963648. (OEIS: A194472)

They are named after Paul Erdős and Jean-Louis Nicolas, who wrote about them in 1975.^[2]

References

^ De Koninck, Jean-Marie (2009). Those Fascinating Numbers. American Mathematical Soc. p. 141. ISBN 978-0-8218-4807-4.
^ Erdős, P.; Nicolas, J.L. (1975), "Répartition des nombres superabondants" (PDF), Bull. Soc. Math. France, 79 (103): 65–90, doi:10.24033/bsmf.1793, Zbl 0306.10025

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[1] De Koninck, Jean-Marie (2009). Those Fascinating Numbers. American Mathematical Soc. p. 141. ISBN 978-0-8218-4807-4.

[2] Erdős, P.; Nicolas, J.L. (1975), "Répartition des nombres superabondants" (PDF), Bull. Soc. Math. France, 79 (103): 65–90, doi:10.24033/bsmf.1793, Zbl 0306.10025

[1]

[2]

v t e Divisibility-based sets of integers
Overview	Integer factorization Divisor Unitary divisor Divisor function Prime factor Fundamental theorem of arithmetic
Factorization forms	Prime Composite Semiprime Pronic Sphenic Square-free Powerful Perfect power Achilles Smooth Regular Rough Unusual
Constrained divisor sums	Perfect Almost perfect Quasiperfect Multiply perfect Hemiperfect Hyperperfect Superperfect Unitary perfect Semiperfect Practical Descartes Erdős–Nicolas
With many divisors	Abundant Primitive abundant Highly abundant Superabundant Colossally abundant Highly composite Superior highly composite Weird
Aliquot sequence-related	Untouchable Amicable (Triple) Sociable Betrothed
Base-dependent	Equidigital Extravagant Frugal Harshad Polydivisible Smith
Other sets	Arithmetic Deficient Friendly Solitary Sublime Harmonic divisor Refactorable Superperfect

See also

References