A212185 Number of exponents >= 2 in canonical prime factorization of n-th highly composite number (A002182(n)).

0, 0, 1, 0, 1, 1, 2, 1, 1, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 2, 2, 2, 2, 3, 2, 2, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 2, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 2, 3, 3, 2, 3, 3, 2, 3, 3, 3, 3, 3, 3, 4, 2, 4, 3, 4, 2

Offset

1,7

Comments

Length of row n of A212184 equals a(n) if a(n) is positive, 1 otherwise.

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

Amiram Eldar, Table of n, a(n) for n = 1..10000

A. Flammenkamp, List of the first 1200 highly composite numbers

S. Ramanujan, Highly Composite Numbers

Example

The canonical prime factorization of 720 (2^4*3^2*5) has 2 exponents that equal or exceed 2. Since 720 = A002182(14), a(14) = 2.

Mathematica

s={}; dm=0; Do[d = DivisorSigma[0, n]; If[d > dm, dm = d; e = FactorInteger[n][[;; , 2]]; AppendTo[s, Count[e, _?(# > 1 &)]]], {n, 1, 10^6}]; s (* Amiram Eldar, Jun 30 2019 *)

Crossrefs

Cf. A002182, A212172, A212182, A212184.

Sequence in context: A347981 A065081 A366643 * A270928 A290257 A196942

Adjacent sequences: A212182 A212183 A212184 * A212186 A212187 A212188

Keyword

nonn

Author

Matthew Vandermast, Jul 16 2012

Status

approved