A practical number is a positive integer positive
integer proper divisors of
For prime factorization prime
numbers and the exponents positive integers ,
a characterization due independently to Stewart (1954) and Sierpiński (1955)
states that iff , for each
where divisor function and the product is defined
to be 1 when
All even perfect numbers are practical. The number
is practical for all A005153 ).
Melfi (1999, p. 726) tabulated 5-tuples of practical numbers of
the form A287683 ).
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References Melfi, G. "On Two Conjectures about Practical Numbers." J. Number Th. 56 , 205-210, 1996. https://doi.org/10.1006/jnth.1996.0012 . Melfi,
G. "On 5-Tuples of Twin Practical Numbers." Boll. Un. Mat. Ital. 2-B ,
723-734, 1999. https://www.bdim.eu/item?id=BUMI_1999_8_2B_3_723_0 . Melfi,
G. "Practical Numbers." http://www.dm.unipi.it/gauss-pages/melfi/public_html/pratica.html . Sierpiński,
W. "Sur une propriété des nombres naturels." Ann. Mat.
Pura Appl. 39 , 69-74, 1955. https://doi.org/10.1007/BF02410762 . Srinivasan,
A. K. "Practical Numbers." Current Sci. 17 , 179-180,
1948. Stewart, B. M. "Sums of Distinct Divisors." Amer.
J. Math. 76 , 779-785, 1954. https://doi.org/10.2307/2372651 . Sloane,
N. J. A. Sequences A005153 /M0991
and A287683 in "The On-Line Encyclopedia
of Integer Sequences." Referenced on Wolfram|Alpha Practical Number
Cite this as:
Weisstein, Eric W. "Practical Number."
From MathWorld https://mathworld.wolfram.com/PracticalNumber.html
Subject classifications
such that every positive
integer 
is the sum of distinct proper divisors of 
. Practical numbers were introduced by
Srinivasan (1948).
with prime factorization 
, where 
are prime
numbers and the exponents 
are positive integers,
a characterization due independently to Stewart (1954) and Sierpiński (1955)
states that 
is practical iff, for each 
, 2, ..., 
,
is the divisor function and the product is defined
to be 1 when 
.
In particular, 
,
so 1 is the only odd practical number.
,
3, .... The first few practical numbers are 1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 28,
30, 32, 36, 40, 42, 48, 54, 56, ... (OEIS A005153).
Melfi (1999, p. 726) tabulated 5-tuples of practical numbers of
the form 
.
The first few centers 
are 18, 30, 198, 306, 462, 1482, 2550, 4422, ... (OEIS A287683).
