Superior Highly Composite Number
المؤلف:
Ramanujan, S.
المصدر:
"Highly Composite Numbers." Proc. London Math. Soc. 14, 347-407, 1915. Reprinted in Collected Papers (Ed. G. H. Hardy et al. ). New York: Chelsea
الجزء والصفحة:
...
1-12-2020
1216
Superior Highly Composite Number
A superior highly composite number is a positive integer 
for which there is an 
such that
for all 
, where the function 
counts the divisors of 
(Ramanujan 1962, pp. 87 and 115). It can be shown that all superior highly composite numbers are highly composite and that the 
th superior highly composite number has the form 
, where the factors 
are prime.
The first few superior highly composite numbers are 2, 6, 12, 60, 120, 360, 2520, 5040, 55440, ... (OEIS A002201), and the corresponding sequence of primes 
is 2, 3, 2, 5, 2, 3, 7, 2, 11, 13, 2, 3, 5, 17, 19, 2, 23, 7, 29, 3, 31, 2, 37, 41, 43, ... (OEIS A000705).
REFERENCES:
Ramanujan, S. "Highly Composite Numbers." Proc. London Math. Soc. 14, 347-407, 1915. Reprinted in Collected Papers (Ed. G. H. Hardy et al. ). New York: Chelsea, pp. 78-129, 1962.
Sloane, N. J. A. Sequences A000705/M0423 and A002201/M1591 in "The On-Line Encyclopedia of Integer Sequences."
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