OFFSET
1,2
COMMENTS
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.
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
From Hal M. Switkay, Aug 27 2023: (Start)
There is a formula defining members of this sequence for all n.
Let the extended natural numbers N+ = {1, 2, 3, ..., oo}, with the ordering 1 < 2 < 3 < ... < oo.
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:
Div+(6) = (1, 2, 3, 6, oo, oo, oo, ...)
Div+(7) = (1, 7, oo, oo, oo, ...)
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.
(End)
It follows from the definition and earlier comments that once a term of this sequence is divisible by lcm({1..k}), all succeeding terms are also divisible by lcm({1..k}). Consequently this sequence is a proper subsequence of A381849. a(n) appears to be divisible by lcm({1..k}) = A003418(k) where k = floor(sqrt(2*n)). - Hal M. Switkay, Sep 07 2025
LINKS
T. D. Noe, Table of n, a(n) for n = 1..448 (terms < 10^100)
Wikipedia, Table of divisors.
FORMULA
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.
EXAMPLE
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.
CROSSREFS
KEYWORD
nonn
AUTHOR
Matthew Vandermast, Jun 09 2004
STATUS
approved
