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A212184 Row n of table gives exponents >= 2 in canonical prime factorization of n-th 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), 347-409; 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

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Last modified October 22 21:09 EDT 2018. Contains 316505 sequences. (Running on oeis4.)