%I A095848
%S A095848 1,2,4,6,12,24,48,60,120,240,360,420,840,1680,2520,5040,10080,15120,
%T A095848 25200,27720,55440,110880,166320,277200,360360,720720,1441440,2162160,
%U A095848 3603600,7207200,10810800,12252240,24504480,36756720,61261200,122522400
%N A095848 Deeply composite numbers: numbers n where sigma_k(n) increases to a record 
               for all sufficiently low values of k.
%C A095848 Sigma_k(n) > sigma_k(m) for all m < n (where the function sigma_k(n) 
               is the sum of the k-th powers of all divisors of n) for all or almost 
               all negative values of k.
%H A095848 Wikipedia, <a href="http://en.wikipedia.org/wiki/Table_of_divisors">Table 
               of divisors</a>.
%F A095848 For n>=4, a(n) is the smallest integer > a(n-1) such that the list of 
               its divisors precedes the list of a(n-1)'s divisors in lexicographic 
               order.
%e A095848 The list of the divisors of a(6)=24, {1,2,3,4,6,8,12,24}, lexicographically 
               precedes the list for the previous term in the sequence (in this 
               case, {1,2,3,4,6,12}, the list for a(5)=12). Therefore 24 belongs 
               in the sequence. 36 does not satisfy this requirement, as {1,2,3,
               4,6,9 . . .} comes after {1,2,3,4,6,8 . . .} in lexicographic order. 
               Since 8^k/9^k increases without limit as k decreases, sigma(36)_k 
               < sigma(24)_k for almost all negative values of k; therefore 36 does 
               not belong in the sequence.
%Y A095848 Cf. A004394, A095849.
%Y A095848 Sequence in context: A048115 A047151 A068010 this_sequence A136339 A019505 
               A135614
%Y A095848 Adjacent sequences: A095845 A095846 A095847 this_sequence A095849 A095850 
               A095851
%K A095848 nonn
%O A095848 1,2
%A A095848 Matthew Vandermast (ghodges14(AT)comcast.net), Jun 09 2004