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A095848 Deeply composite numbers: numbers n where sigma_k(n) increases to a record for all sufficiently low (i.e., negative) values of k.

%I A095848 #24 Sep 27 2023 13:19:02
%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 (i.e., negative) 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.
%C A095848 This sequence is infinite, because it includes every term in A051451. This follows from the formula for a(n), and the fact that A051451 consists of the distinct terms of A003418. - _Hal M. Switkay_, Mar 22 2021
%C A095848 From _Hal M. Switkay_, Aug 27 2023: (Start)
%C A095848 There is a formula defining members of this sequence for all n.
%C A095848 Let the extended natural numbers N+ = {1, 2, 3, ..., oo}, with the ordering 1 < 2 < 3 < ... < oo.
%C A095848 For every natural number k, let Div+(k) be an infinitely long vector of extended natural numbers, starting with the divisors of k in increasing order, followed by infinitely many coordinates equal to oo. For example:
%C A095848   Div+(6) = (1, 2, 3, 6, oo, oo, oo, ...)
%C A095848   Div+(7) = (1, 7, oo, oo, oo, ...)
%C A095848 Then for all natural numbers n, a(n) = k if and only if k is the smallest natural number such that Div+(k) lexicographically precedes Div+(a(i)), for 1 <= i < n.
%C A095848 (End)
%H A095848 T. D. Noe, <a href="/A095848/b095848.txt">Table of n, a(n) for n = 1..448</a> (terms < 10^100)
%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.
%e A095848 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 Cf. A003418, A051451.
%K A095848 nonn
%O A095848 1,2
%A A095848 _Matthew Vandermast_, Jun 09 2004