TOPICS
Deficiency
Given binomial coefficient 
, write
 |
for 
, where 
contains only those prime factors 
. Then the number of 
for which 
(i.e., for which all the factors of 
are 
is called the deficiency of 
(Erdős et al. 1993, Guy 1994). The following
table gives the good binomial coefficients
(i.e., those with 
)
having deficiency 
(Erdős et al. 1993), and Erdős et al. (1993) conjecture that
there are no other with 
.
 | good binomial coefficients |
| 1 |  ,  ,  ,  ,  ,  ,  , ... |
| 2 |  ,
 ,  ,  ,  ,  , |
 ,  | |
| 3 |  ,
 ,  ,  ,  ,  |
| 4 |  |
| 9 |  |
See also
Abundance, Good Binomial CoefficientExplore with Wolfram|Alpha
References
Erdős, P.; Lacampagne, C. B.; and Selfridge, J. L. "Estimates of the Least Prime Factor of a Binomial Coefficient." Math. Comput. 61, 215-224, 1993.Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 84-85, 1994.Referenced on Wolfram|Alpha
DeficiencyCite this as:
Weisstein, Eric W. "Deficiency." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Deficiency.html