

A212184


Row n of table gives exponents >= 2 in canonical prime factorization of nth highly composite number (A002182(n)), in nonincreasing order, or 0 if no such exponent exists.


1



0, 0, 2, 0, 2, 3, 2, 2, 4, 2, 3, 2, 2, 4, 3, 2, 4, 2, 3, 2, 2, 4, 3, 2, 4, 2, 3, 3, 5, 2, 4, 3, 6, 2, 4, 2, 2, 3, 2, 4, 4, 5, 2, 2, 4, 2, 3, 3, 5, 2, 4, 3, 6, 2, 4, 2, 2, 5, 3, 4, 4, 5, 2, 2, 6, 3, 4, 2, 3, 3, 5, 2, 4, 3, 6, 2, 4, 2, 2, 5, 3, 4, 4, 5, 2, 2, 6
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,3


COMMENTS

Length of row n equals A212185(n) if A212185(n) is positive, or 1 if A212185(n) = 0.
Row n of table represents second signature of A002182(n) (cf. A212172). The use of 0 in the table to represent squarefree highly composite numbers accords with the usual OEIS practice of using 0 to represent nonexistent elements when possible. In comments, the second signature of squarefree numbers is represented as { }.
No row is repeated an infinite number of times in the table. The contrary to this would imply that at least one integer appeared in A212183 an infinite number of times  something that Ramanujan proved to be false (cf. Ramanujan link). It would be interesting to know if there is an upper bound on the number of times a row can appear.


REFERENCES

S. Ramanujan, Highly composite numbers, Proc. Lond. Math. Soc. 14 (1915), 347409; reprinted in Collected Papers, Ed. G. H. Hardy et al., Cambridge 1927; Chelsea, NY, 1962.


LINKS

Table of n, a(n) for n=1..87.
A. Flammenkamp, List of the first 1200 highly composite numbers
S. Ramanujan, Highly Composite Numbers (p. 34)


FORMULA

Row n is identical to row A002182(n) of table A212172.


EXAMPLE

First rows read: 0; 0; 2; 0; 2; 3; 2,2; 4; 2; 3; 2,2; 4;...
12 = 2^2*3 has positive exponents 2 and 1 in its canonical prime factorization (1s are often left implicit as exponents). Only exponents that are 2 or greater appear in a number's second signature; therefore, 12's second signature is {2}. Since 12 = A002182(5), row 5 represents the second signature {2}.


CROSSREFS

Cf. A002182, A212172, A212182, A212183, A212185.
Sequence in context: A208295 A285721 A214292 * A033769 A074660 A002125
Adjacent sequences: A212181 A212182 A212183 * A212185 A212186 A212187


KEYWORD

nonn,tabf


AUTHOR

Matthew Vandermast, Jul 01 2012


STATUS

approved



