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

1, 2, 4, 6, 12, 24, 48, 60, 120, 240, 360, 420, 840, 1680, 2520, 5040, 10080, 15120, 25200, 27720, 55440, 110880, 166320, 277200, 360360, 720720, 1441440, 2162160, 3603600, 7207200, 10810800, 12252240, 24504480, 36756720, 61261200, 122522400

Offset: 1

Author

Matthew Vandermast, Jun 09 2004

Keywords

• nonn

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)

Examples

			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.
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..448 (terms < 10^100)
• Wikipedia, Table of divisors.

Crossrefs

Cf. A004394, A095849.

Cf. A003418, A051451.

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.