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

Author

Matthew Vandermast, Jul 16 2012

Keywords

• nonn

Comments

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

Examples

			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.
		

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

Crossrefs

Cf. A002182, A212172, A212182, A212184.

Programs

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 *)