A212183 Largest odd divisor of A002183(n) (number of divisors of n-th highly composite number).

1, 1, 3, 1, 3, 1, 9, 5, 3, 1, 9, 5, 3, 15, 1, 9, 5, 3, 15, 1, 9, 5, 21, 45, 3, 25, 27, 15, 1, 9, 5, 21, 45, 3, 25, 27, 7, 15, 1, 9, 5, 21, 45, 3, 25, 27, 7, 15, 63, 1, 9, 75, 5, 21, 45, 3, 25, 27, 7, 15, 63, 1, 9, 75, 5, 21, 45, 3, 25, 105, 27, 7, 15, 63, 1, 9

Offset

1,3

Comments

The "odd part" (largest odd divisor) of the number of divisors of n is a function of the exponents >=2 in the prime factorization of n (cf. A212172, A212181).

The number 1 appears a total of 18 times (see Graeme link for proof). Ramanujan proved that no number appears an infinite number of times (see Ramanujan link). It would be interesting to know more about a) which odd numbers appear in the sequence and b) how many times a number of a given size can appear in the sequence. See also A160233.

Links

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

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

Graeme McRae, Highly Composite Numbers

S. Ramanujan, Highly Composite Numbers (p. 34).

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.

Formula

a(n) = A000265(A002183(n)) = A212181(A002182(n)).

Example

The highly composite number 48 has a total of 10 divisors. Since 48 = A002182(8), A002183(8) = 10. Since the largest odd divisor of 10 is 5, a(8) = 5.

Crossrefs

Cf. A000265, A002182, A002183, A212181.

A160233 gives the n-th integer that is the largest member of A002183 with its particular odd part.

Sequence in context: A073575 A363533 A146431 * A115716 A079412 A356655

Adjacent sequences: A212180 A212181 A212182 * A212184 A212185 A212186

Keyword

nonn

Author

Matthew Vandermast, Jun 08 2012

Status

approved