cp's OEIS Frontend

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

%I A212183 #16 Jul 01 2019 11:14:14
%S A212183 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,
%T A212183 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,
%U A212183 27,7,15,63,1,9,75,5,21,45,3,25,105,27,7,15,63,1,9
%N A212183 Largest odd divisor of A002183(n) (number of divisors of n-th highly composite number).
%C A212183 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).
%C A212183 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.
%H A212183 Amiram Eldar, <a href="/A212183/b212183.txt">Table of n, a(n) for n = 1..10000</a>
%H A212183 A. Flammenkamp, <a href="http://wwwhomes.uni-bielefeld.de/achim/highly.txt">List of the first 1200 highly composite numbers</a>
%H A212183 Graeme McRae, <a href="http://2000clicks.com/mathhelp/NumberFactorsHighlyComposite.aspx">Highly Composite Numbers</a>
%H A212183 S. Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/Cpaper15/page34.htm">Highly Composite Numbers</a> (p. 34).
%H A212183 S. Ramanujan, <a href="https://doi.org/10.1112/plms/s2_14.1.347">Highly composite numbers</a>, Proc. Lond. Math. Soc. 14 (1915), 347-409; reprinted in Collected Papers, Ed. G. H. Hardy et al., Cambridge 1927; Chelsea, NY, 1962.
%F A212183 a(n) = A000265(A002183(n)) = A212181(A002182(n)).
%e A212183 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.
%Y A212183 Cf. A000265, A002182, A002183, A212181.
%Y A212183 A160233 gives the n-th integer that is the largest member of A002183 with its particular odd part.
%K A212183 nonn
%O A212183 1,3
%A A212183 _Matthew Vandermast_, Jun 08 2012